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H?"  eUFL, January 2004    Hঘ?"  fM. Figueiredo, IST rB   < Dp?"<  c $޽h ? ̙33 Default Design 0U(    <tn7% ;Feature Selection for Supervised and Unsupervised Learning<<$ < g  <nS  Mrio A. T. Figueiredo Institute of Telecommunications, and Instituto Superior Tcnico Technical University of Lisbon PORTUGAL>%(   Hn?"  z4Work herein reported was done in collaboration with:555  Htn?"1r oL. Carin, B. Krishnapuram, and A. Hartemink, Duke University; A. K. Jain and M. Law, Michigan State University.bp  p   H0n?"   *mtf@lx.it.pt www.lx.it.pt/~mtf$ U 0 Z  Bd޽h @ ? ̙33y___PPT10Y+D=' = @B +  @ (     <nlo ]Outline   < ^`p$ 0<4___PPT9 9Introduction Review LASSO Regression The LASSO Penalty for Multinomial Logistic Regression Bound Optimization Algorithms (Parallel and Sequential) Non-linear Feature Weighting/Selection Experimental Results  @`  B* r ,$ 0 pPart I Supervised Learning   Bp/ w r8 ,$ 0 t Part III Unsupervised Learning!!    </  ^bp$ 0<4___PPT9 Introduction Review of Model-Based Clustering with Finite Mixtures Feature Saliency An EM Algorithm to Estimate Feature Saliency Model Selection Experimental Results    @`  <3 r 9 ,$ 0 zPart II Performance Bounds  H  0޽h ? ̙33LD___PPT10$+OX=D' = @B DS' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(+p+0+   ++0+   +( ((P>p ((    <tI hE  kSupervised Learning    BdK 8 )Goal: to learn a functional dependency...$* ) d  <8cP ^B  6D8c.^B  6D8c t # # | A?txp_fig"84 SOURCE\documentclass{slides} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ {\bf x}\in \mathbb{R}^d \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION300*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono.ORIGWIDTH 61,8754PICTUREFILESIZE716  B`] ' d....from a set of examples (the training data):.  * # | A?txp_fig". 2* SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ {\cal D} = \left\{ ({\bf x}_1,y_1), ({\bf x}_2,y_2), \dots, ({\bf x}_n,y_n)\right\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1566PICTUREFILESIZE2557 4 # | A8c?txp_fig"  ; SOURCEl\documentclass{slides}\pagestyle{empty} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \begin{document} \[ f(\cdot,\bbeta ) \] \end{document} \begin{document} \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseJ2WORKAROUNDTRANSPARENCYBUGTrueD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH596PICTUREFILESIZE3633 6 # | A8c?txp_fig"@X SOURCE<\documentclass{article}[12pt]\pagestyle{empty} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \begin{document} \[ y = f({\bf x},\bbeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseJ2WORKAROUNDTRANSPARENCYBUGTrueD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH496PICTUREFILESIZE23296 8 BTs ~   X Discriminative (non-generative) approach: no attempt to model the joint density :Y-:YR ; # | A?txp_fig"i   SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ p({\bf x},y)\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH284PICTUREFILESIZE775XB < 0D8c"  = H ?"< @set of parameters> >  v A?txp_fig"'9 SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ k \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono,ORIGWIDTH 4,8754PICTUREFILESIZE292H  0޽h ? ̙337 t1l1`-0 0(    < `PN .Complexity Control Via Bayes Point Estimation//. @8 $u +u$  B $3  \&Bayesian (point estimation) approach: &   B  2Q&q ;Prior    B T Qq Pcontrols the  complexity of ( fB   6D8c1    <( +u c1Good generalization requires complexity control.1   v A ?txp_fig"qXH SOURCE,\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ f(\cdot,\mbox{\boldmath$\beta$}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH274PICTUREFILESIZE723~   v A?txp_fig"PN< SOURCE \documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ p(\mbox{\boldmath$\beta$}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono.ORIGWIDTH 19,8754PICTUREFILESIZE677l (3  ,3( ,$D 0  B0 13S JLikelihood function fB  6D8c    BX (  b,Maximum a posteriori (MAP) point estimate of,    v A_?txp_fig"I ?N SOURCE2\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ p({\cal D}|\mbox{\boldmath$\beta$}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH304PICTUREFILESIZE880n   v A?txp_fig" 6 SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \mbox{\boldmath$\beta$} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE348\ " # | A1?txp_fig"+d   SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \widehat{\mbox{\boldmath$\beta$}} = \arg\max_{\mbox{\boldmath$\beta$}} \; p(\mbox{\boldmath$\beta$})p({\cal D}|\mbox{\boldmath$\beta$}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1086PICTUREFILESIZE2649t l  / - /,$D 0 # < P p pHPrediction for a new  input :  %  )  v A?txp_fig"v /z SOURCE^\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \widehat{y} = f({\bf x},\widehat{\mbox{\boldmath$\beta$}}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH496PICTUREFILESIZE1179N *  v A?txp_fig") ) SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ {\bf x} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE269H  0޽h ? ̙33___PPT10+PD~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-%(+ p  >(    <0 p0$ 3Bayes Point Estimate Versus Fully Bayesian Approach443   <# P  |TPoint prediction for a new  input :  +    v A?txp_fig"z z SOURCE^\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \widehat{y} = f({\bf x},\widehat{\mbox{\boldmath$\beta$}}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH496PICTUREFILESIZE1179F   v A?txp_fig"'  SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ {\bf x} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE269z   v A?txp_fig"7IJ SOURCE.\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \widehat{ \mbox{\boldmath$\beta$}} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE404  < 9 P7 EPoint estimate     BX<  ^  5We will not consider fully Bayesian approaches here. $6+5   BTB s  PFully Bayesian prediction  :  # | A?txp_fig"" x SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \widehat{y} = \int f({\bf x},\mbox{\boldmath$\beta$})\, p(\mbox{\boldmath$\beta$}|{\cal D})\, d\mbox{\boldmath$\beta$} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono0ORIGWIDTH106,8756PICTUREFILESIZE2598  NN ?" s  1 H  0޽h ? ̙33' ##9Pi#(    <R l0  Linear (w.r.t. b ) Regression6    6Y ?"L #  sAe.g., radial basis functions, splines, wavelets, polynomials,...A & 6] ?"p.C e3We consider functions which are linear w.r.t. 3 ' 6@b ?"S 8where  ( 6|f ?"  S!is some dictionary of functions; ! + # | A?txp_fig" d SOURCEH\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \{ \psi_1 ({\bf x}),\dots , \psi_k ({\bf x})\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH816PICTUREFILESIZE1606l . # | A?txp_fig"I6 SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \mbox{\boldmath$\beta$} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE345 2 # | A?txp_fig";B SOURCE&\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ f({\bf x},\mbox{\boldmath$\beta$}) = \beta_0 + \sum_{i=1}^k \beta_i \psi_i ({\bf x}) = \beta_0 + \mbox{\boldmath$\beta$}^T \mbox{\boldmath$\psi$}({\bf x}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono0ORIGWIDTH184,8756PICTUREFILESIZE44031 l CJ   9J C ,$D 0 / 0$ ?"CJ |j  ^,Notable particular cases: linear regression:,j 3  v A?txp_fig"Z It 0 SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \psi_i ({\bf x}) = x_i\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH464PICTUREFILESIZE974 4 0 ?" i  [)kernel regression: as in SVM, RVM, etc...) 7  v A?txp_fig"  R SOURCE6\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \psi_i ({\bf x}) = K({\bf x},{\bf x}_i)\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH736PICTUREFILESIZE1532H  0޽h ? ̙33~___PPT10^+IDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*9%(+( $$F t$(    < p0  x"Likelihood Function for Regression##"   Hp ?""B g5Likelihood function, for Gaussian observation model 5  H ?"] 7whereX (  v A?txp_fig"H*" SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ p({\cal D}|\mbox{\boldmath$\beta$}) \propto \exp\left\{ -\frac{1}{2\sigma^2}\, \|\beta_0+{\bf H}\mbox{\boldmath$\beta$}-{\bf y}\|_2^2\right\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1746PICTUREFILESIZE3740F - Nԭ ?" l  JAssuming that y and the columns of H are centered, we drop w.l.o.g.6K'JJ / # | A?txp_fig"f  SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \beta_0 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH104PICTUREFILESIZE412:l   E ,$D 0 7 H ?"   f4Maximum likelihood / ordinary least squares estimate4 ? H| ?"] V$& undetermined if r A  v A?txp_fig"&y 8 SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \mbox{rank}({\bf H}) < k \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH554PICTUREFILESIZE990\ B  v A?txp_fig"  SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \widehat{\mbox{\boldmath$\beta$}}_{\mbox{\tiny OLS}} = \arg\min_{\mbox{\boldmath$\beta$}}\|{\bf H}\mbox{\boldmath$\beta$}-{\bf y}\|_2^2 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono0ORIGWIDTH115,8756PICTUREFILESIZE2119r D  v A?txp_fig" > SOURCE"\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ H_{j,l} = \psi_l ({\bf x}_j) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH576PICTUREFILESIZE1188 F N 8c?"*3J  N&& the design matrix.  H  0޽h ? ̙33~___PPT10^+LDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*E%(+/ !)),((    <d Hp Ccalled ridge regression, or weight decay (in neural nets parlance).$D-C   < (EH MWith a Gaussian prior    <P k: /Bayesian Point Estimates: Ridge and the LASSO 00/    v A`?txp_fig"S  SOURCEt\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \widehat{\mbox{\boldmath$\beta$}}_{\mbox{\tiny Ridge}} = \arg\min_{\mbox{\boldmath$\beta$}}\left\{ \|{\bf H}\mbox{\boldmath$\beta$}-{\bf y}\|_2^2 + \lambda \|\mbox{\boldmath$\beta$}\|_2^2\right\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1716PICTUREFILESIZE3274 ' < ,$ 0 LASSO regression [Tibshirani, 1996], prunning priors for NNs [Williams, 1995], basis pursuit [Chen, Donoho, Saunders, 1995].4_  )  v A?txp_fig"+gLD SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ p(\mbox{\boldmath$\beta$})\propto \exp\left\{-\tau \|\mbox{\boldmath$\beta$}\|_2^2\right\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1146PICTUREFILESIZE25066l M-  ,M- ,$D 0  </ M  OWith a Laplacian prior      v AX?txp_fig"( PH SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ p(\mbox{\boldmath$\beta$})\propto \exp\left\{-\tau \|\mbox{\boldmath$\beta$}\|_1\right\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono0ORIGWIDTH112,8756PICTUREFILESIZE2109 "  v At?txp_fig"y_ _|  SOURCEp\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \widehat{\mbox{\boldmath$\beta$}}_{\mbox{\tiny LASSO}} = \arg\min_{\mbox{\boldmath$\beta$}}\left\{ \|{\bf H}\mbox{\boldmath$\beta$}-{\bf y}\|_2^2 + \lambda \|\mbox{\boldmath$\beta$}\|_1\right\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono0ORIGWIDTH175,8756PICTUREFILESIZE3193 # H/ ?"  Qpromotes sparseness of  %  v Aw?txp_fig" p SOURCET\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \widehat{\mbox{\boldmath$\beta$}}_{\mbox{\tiny LASSO}} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono.ORIGWIDTH 28,8754PICTUREFILESIZE745 & HT#/ ?" -  uCi.e., its components are either significantly large, or zero. C`B * 0D8c 2 2  + H('/ ?"   _feature selectionH  0޽h ? ̙33___PPT10+ nD' = @B DF' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(+8+0+'/  +5 ?" j !B  B D8c?"z X B  B D8c?" U 3B  B D8c?" Y Y f  # | A ?txp_fig"c ( SOURCE \documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \frac{\lambda}{2} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE434j  # | A ?txp_fig" 7 * SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ -\frac{\lambda}{2} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH154PICTUREFILESIZE471  # | Aw?txp_fig"\Gi t SOURCEX\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \widehat{\mbox{\boldmath$\beta$}}_{\mbox{\tiny LASSO}} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono.ORIGWIDTH 28,8754PICTUREFILESIZE745K  H|s/ ?" & ,$ 0 3More insight on the LASSO, see [Tibshirani, 1996]. $43"  Ty/ 8c?"  .[Efron, Hastie, Johnstone, & Tibshirani, 2002]//*,   N~/ 8c?"  b:Least angle regression (LAR); currently the best approach. ; H  0޽h ? ̙33___PPT10+#D' = @B DF' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+/  +C m:e:(9(  ( ( <t/ g:c  EM Algorithm for the LASSO(  ( T\/ 8c?""u8 e=LASSO can be computed by EM using a hierarchical formulation: >  (  v A8c?txp_fig"5n SOURCER\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ p(\beta_i|\tau_i)={\cal N}(\beta_i|0,\tau_i) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH368(BOXHEIGHT321"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1066PICTUREFILESIZE5930 ( N/ 8c?"A3a > are independent,   V  (  v A.8c?txp_fig"fA  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \beta_i\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH368(BOXHEIGHT321"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH104PICTUREFILESIZE860  (  v A28c?txp_fig"'d  SOURCEr\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ p(\tau_i) \propto \exp\left\{-\frac{\gamma}{2}\tau_i\right\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH368(BOXHEIGHT321"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1026PICTUREFILESIZE6121 l eR (eR,$D 0B  ( B D8c?"f  ( # | A38c?txp_fig"B R> SOURCE"\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \begin{eqnarray*} p(\beta_i)& = & \int {\cal N}(\beta_i|0,\tau_i)p(\tau_i)d\tau_i \\ &\propto& \exp\left\{-\sqrt{\gamma}|\beta_i|\right\} \end{eqnarray*} \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH368(BOXHEIGHT321"BOXFONT10(BOXWRAP FalseJ2WORKAROUNDTRANSPARENCYBUGTrueD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1618PICTUREFILESIZE 14623zB ( < D8c?"ee ( N/ 8c?"Oo,$D 0 This possibility mentioned in [Tibshirani, 1996]; not very efficient, but& $K&   " :l  (,$D 09 ( T/ 8c?". ,$D 0 kCTreat the as missing data and apply standard EM. This leads to: D  ( # | A08c?txp_fig"F $D 0" SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \tau_i \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH368(BOXHEIGHT321"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE4972 ( # | A48c?txp_fig"   p$D 0< SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\bf V}^{(t)} = \gamma\,\mbox{diag}\{|\beta_k^{(t)}|^{-1},...,|\beta_k^{(t)}|^{-1}\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH368(BOXHEIGHT321"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1696PICTUREFILESIZE8042U ( T/ 8c?" ,$D 0 Ewhich can be called an iteratively reweighted ridge regression (IRRR)&#    N (  v A8c?txp_fig" ] $D 0^ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}^{(t+1)} = \left(\sigma^2{\bf V}^{(t)} +{\bf H}^T{\bf H}\right)^{-1}{\bf H}^T{\bf y} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH368(BOXHEIGHT321"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1726PICTUREFILESIZE7268H ( 0޽h ? ̙33C;___PPT10+tD' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(+8+0+(/  +, 0$($,#(  , , </ g:. mAbout g( 8 ,Q= ,Q,= , N0/ 8c?",Qq _7The previous derivation opens the door to other priors. 8  , N/ 8c?"V  `For example, a Jeffreys prior:&    ,  v A58c?txp_fig" =Z SOURCE>\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ p(\tau_i) \propto \frac{1}{\tau_i} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH368(BOXHEIGHT321"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH516PICTUREFILESIZE2911l ]V  ,V] ,$D 0  , N+ 8c?"]Vv S+The EM algorithm becomes (still IRRR-type): ,   ,  v A78c?txp_fig"x   SOURCE~\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\bf U}^{(t)} = \mbox{diag}\{|\beta_k^{(t)}|,...,|\beta_k^{(t)}|\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH368(BOXHEIGHT321"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1386PICTUREFILESIZE6106  ,  v A88c?txp_fig"H SOURCE,\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}^{(t+1)} = {\bf U}^{(t)}\left(\sigma^2{\bf I} +{\bf U}^{(t)}{\bf H}^T{\bf H}{\bf U}^{(t)}\right)^{-1} {\bf U}^{(t)}{\bf H}^{(t)}{\bf y} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH368(BOXHEIGHT321"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2406PICTUREFILESIZE9253{l N T  ,N T ,$D 0 , N@+ 8c?"N T  Interestingly, similar to the FOCUSS algorithm for regression with an penalty [Kreutz-Delgado & Rao, 1998]. Strong sparseness!$S@T      X ,  v A68c?txp_fig"g G  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ l_0 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH368(BOXHEIGHT321"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE544b , N&+ 8c?" L ,$D 0 zProblem: non-convex objective, results depend on initialization. Possibility: initialize with OLS estimate. { H , 0޽h ? ̙33C;___PPT10+doD' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(+8+0+,+  +;& ]"U" 0!(  0 0 <D1+ g: r Some Results(    0 N(7+ 8c?"3 9 *Same vectors as in [Tibshirani, 1996]:+&     X 0 # | AV8c?txp_fig"cU#" SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bbeta \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH368(BOXHEIGHT321"BOXFONT10(BOXWRAP FalseJ2WORKAROUNDTRANSPARENCYBUGTrueD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE713 0 # | A8c?txp_fig"q Z SOURCE>\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bbeta^{(a)} = [3,1.5,0,0,2,0,0,0] \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH368(BOXHEIGHT321"BOXFONT10(BOXWRAP FalseJ2WORKAROUNDTRANSPARENCYBUGTrueD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1356PICTUREFILESIZE5612  0 # | A8c?txp_fig" aV SOURCE:\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bbeta^{(b)} = [5,0,0,0,0,0,0,0] \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH368(BOXHEIGHT321"BOXFONT10(BOXWRAP FalseJ2WORKAROUNDTRANSPARENCYBUGTrueD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1256PICTUREFILESIZE4735>  0 NS+ 8c?"Iui DDesign matrices and experimental procedure as in [Tibshirani, 1996] $E1&2      l u 0u,$D 0 0 B A8c?"& P  0 N\[+ 8c?"A Y1Model error (ME) improvement w.r.t. OLS estimate: 2  0  v A8c?txp_fig"c z SOURCE^\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ ME =E[\|{\bf H}\widehat{\bbeta}-{\bf H}\bbeta\|^2] \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH368(BOXHEIGHT321"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1226PICTUREFILESIZE4485 0 Nf+ 8c?"o #u |Close to best in each case, without any cross-validation; more results in [Figueiredo, NIPS 2001] and [Figueiredo, PAMI 2003].J5@K       H 0 0޽h ? ̙33~___PPT10^+DB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(+C ::A:(    <w+ k:  ~(Classification via Logistic Regression ))(   B~+ >^ ABinary classification: f   v A ?txp_fig"g _, SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{\bf x} \pagestyle{empty} \begin{document} \[ y\in\{-1,+1\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono.ORIGWIDTH 67,8754PICTUREFILESIZE904l }  ?} ,$D 0  B+ }  tJMulti-class, with  1 of m encoding: &   v A?txp_fig"v cD SOURCE(\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{\bf x} \pagestyle{empty} \begin{document} \[ y=[y^{(1)},\dots,y^{(m)}] \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH936PICTUREFILESIZE1543 ! B+  ] }  /Class  $  v A?txp_fig"   d\ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{\bf x} \pagestyle{empty} \begin{document} \[ i \Leftrightarrow \left\{\begin{array}{l}y^{(i)}=1\\ y^{(l)}=0 \Leftarrow l\neq i\end{array}\right. \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono0ORIGWIDTH115,8756PICTUREFILESIZE2740Wl ~ >~,$D 0 2 B 8c?#" `~ 3 6+ Q kRecall that may denote the components of , or other (nonlinear) functions of , such as kernels. lll~ 5  v A?txp_fig"ue3D SOURCE(\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ \mbox{\boldmath$\psi$}({\bf x}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH224PICTUREFILESIZE642L ;  v A?txp_fig"oT SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ {\bf x} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE269L =  v A?txp_fig"U~ SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ {\bf x} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE269 @  v A?txp_fig"G^ SOURCEf\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\bpsi}{\mbox{\boldmath$\psi$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ P \left( y=1|\xv{},\bbeta \right) = \frac{\exp \left( \bbeta^T \bpsi({\bf x}) \right)}{1 + \exp \left( \bbeta^T \bpsi({\bf x}) \right)} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1818PICTUREFILESIZE 13855@ A  v A?txp_fig"E y$D 0X SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\bpsi}{\mbox{\boldmath$\psi$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ P \left( y^{(i)}=1|\xv,\bbeta \right) = \frac{\exp \left( (\bbeta^{(i)})^T \bpsi({\bf x}) \right)}{\displaystyle \sum_{l=1}^m \exp \left( (\bbeta^{(l)})^T \bpsi({\bf x}) \right)} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2148PICTUREFILESIZE 21838H  0޽h ? ̙33>6___PPT10+G'D' = @B D' = @BA?%,( < +O%,( < +Dt' =%(D' =%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*>%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*A%(+L EE TE(  T   <+ k:  ~(Classification via Logistic Regression ))( l   T  ,$D 0   B+ ! /Since    v A?txp_fig"q e~ SOURCEb\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \sum_{i=1}^m P \left( y^{(i)}=1|\xv,\bbeta \right) = 1\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1246PICTUREFILESIZE7332   BH+   T*, we can set w.l.o.g. +p    v A?txp_fig" 6 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bbeta^{(m)} = 0 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH454PICTUREFILESIZE917   6+   AParameters to estimate:    v A?txp_fig"y% j SOURCEN\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bbeta = [\bbeta^{(1)},...,\bbeta^{(m-1)}] \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1036PICTUREFILESIZE1751 T  v A?txp_fig" _[A`X SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\bpsi}{\mbox{\boldmath$\psi$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ P \left( y^{(i)}=1|\xv,\bbeta \right) = \frac{\exp \left( (\bbeta^{(i)})^T \bpsi({\bf x}) \right)}{\displaystyle \sum_{l=1}^m \exp \left( (\bbeta^{(l)})^T \bpsi({\bf x}) \right)} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2148PICTUREFILESIZE 21838%l    T ,$D 0dt [  T $,$D 0   6 8 [ q{  J Maximum log-likelihood estimate:! !   v A!?txp_fig"b m  (  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}_{\mbox{\tiny ML}}= \arg\max_{\bbeta}\,\ell(\bbeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono0ORIGWIDTH100,8756PICTUREFILESIZE25204 T  v A?txp_fig"< xp SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\bpsi}{\mbox{\boldmath$\psi$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \ell(\bbeta) = \! \sum_{j=1}^{n} \left[ \sum_{i=1}^{m} y_j^{(i)} {(\bbeta^{(i)}})^T \bpsi({\bf x}_j) - \log\! \sum_{i=1}^{m} \exp\left( {(\bbeta^{(i)}})^T \bpsi({\bf x}_j) \! \right) \right] \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH3118PICTUREFILESIZE 23595N ^n T n T 6"8 ^n oEIf is separable, is unbounded, thus undefined.Fj  T  v A?txp_fig"( 0 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \ell(\bbeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH224PICTUREFILESIZE814\  T  v A ?txp_fig"M" SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\cal D}\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH104PICTUREFILESIZE379  T  v A?txp_fig"V SOURCE:\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}_{\mbox{\tiny ML}}\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH206PICTUREFILESIZE1344H T 0޽h ? ̙33___PPT10.P+KD~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* T%(+) %% P*%(  P  <+ k: -Penalized (point Bayes) Logistic Regression ..- rl 3a  P$`,$ 0  6O8 3a  Q'Penalized (or point Bayes MAP) estimate(  6S8 _  '  /where P  v A?txp_fig"   SOURCE~\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}_{\mbox{\tiny MAP}}= \arg\max_{\bbeta}\,L(\bbeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1116PICTUREFILESIZE6694 P  v A?txp_fig"` I  h SOURCEL\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ L(\bbeta) = \ell(\bbeta) + \log p(\bbeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1186PICTUREFILESIZE6532O l     P-OY,$ 0  6 g8    8Gaussian prior`B  0D8cH H  !  v A]?txp_fig" d SOURCEH\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \log p(\bbeta) = -\lambda\|\bbeta\|_2^2 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH976PICTUREFILESIZE4761`B P 0D8cjG kG   P 6s8    =Penalized log. reg.* l Yy x  Py Yx,$D 0t t 3 ;  PYy  ,$D 0`B  0D8c    6x8 5 ; GLaplacian prior (LASSO prior)`B  0D8ck l   6T|8 3 9 O%favors sparseness, feature selection.& "  v A^?txp_fig" O` SOURCED\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \log p(\bbeta) = -\lambda\|\bbeta\|_1 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH976PICTUREFILESIZE42222  P N8 8c?" r x,$D 0 jBFor linear regression it does, what about for logistic regression? C H P 0޽h ? ̙33___PPT10n.P&$+iUDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* P%(+ {s =Tp(    <8 k> )Laplacian Prior for Logistic Regression **)   HT8 ?"*t J M#Simple test with 2 training points:$o,8 ~J (  1~J ( B   H DV?"88B   N DV?"^ ^2  < ?#" `P  B 8c?#" `B  B D8c?"*:B  B D8c?"DT  H8 ?"l 1class 1   H08 ?"s(  2class -1 V "  v A%?txp_fig"7 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ d \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE537Z $  v A'?txp_fig"aKC SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ -d \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH154PICTUREFILESIZE612B % B D8c?"pp\B & B D8c?"^V )  v A(?txp_fig"Q  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ 1 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH54PICTUREFILESIZE222Z +  v A)?txp_fig"nJ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ -1 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH144PICTUREFILESIZE265\ .  v A+?txp_fig"J   SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ x_1 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH114PICTUREFILESIZE603\ 0  v A,?txp_fig"h~   SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ x_2 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH114PICTUREFILESIZE777P"l < i Ti< ,$D 0z ; < A??"< Xv @  v A/?txp_fig"  < SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}_2 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH134PICTUREFILESIZE586V A  v A%?txp_fig"0$   SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ d \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE537 D H8 ?" ic  Xw/ Laplacian prior\ J  v A:?txp_fig"A  " SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \tau \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH64PICTUREFILESIZE313` L  v A*?txp_fig"0v Z $ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ -\tau \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH154PICTUREFILESIZE371 `  v AY?txp_fig" ~ SOURCEb\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \tau = \frac{\lambda(1+e^{\widehat{\beta}_1})}{2} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH756PICTUREFILESIZE3726   v A&?txp_fig"T >p SOURCET\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\cal D} = \{ ([1,d]^T,1),([-1,-d]^T,-1) \} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1716PICTUREFILESIZE6077   v A?txp_fig" ,$ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\bpsi}{\mbox{\boldmath$\psi$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bbeta = [\beta_1,\;\beta_2]^T \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH686PICTUREFILESIZE3125 L  v A\?txp_fig" ,$ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\bpsi}{\mbox{\boldmath$\psi$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bpsi({\bf x}) = [x_1,\;x_2]^T \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH866PICTUREFILESIZE3851;l T|  Q|T ,$D 0e-T 4  9# T|L  z : 6 ?#" `< ,T 48  ;# 48 B < B DV?" T 8 B = H DV?"  2 > < ?#" ` . p ? B 8c?#" `a  B @ B D8c?" B A B D8c?" R R B H%: ?" 13 1class 1 C Hx): ?"  2class -1 V D  v A%?txp_fig"   SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ d \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE537Z E  v A'?txp_fig" u   SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ -d \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH154PICTUREFILESIZE612B F B D8c?" h  V G  v A(?txp_fig" 7   SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ 1 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH54PICTUREFILESIZE222Z H  v A)?txp_fig" t  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ -1 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH144PICTUREFILESIZE265\ I  v A,?txp_fig" 4   SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ x_2 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH114PICTUREFILESIZE777\ J  v A+?txp_fig"H    SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ x_1 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH114PICTUREFILESIZE603B K B D8c?"{ { @ / , P/ ,\ 4 # | A%?txp_fig" 7  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ d \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE537 2 Nlf: ?"/ , X.As decreases, becomes less relevant/\ M  v A,?txp_fig"jo   SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ x_2 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH114PICTUREFILESIZE777 N H8r: ?"  DLinear logistic regressionl I  SI  ,$D 0 < B A=?"l V B  v A%?txp_fig"@ 0  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ d \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE537 C N|: ?"PI 4 C  Ww/ Gaussian priorv ?  v A/?txp_fig"Cg g < SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}_2 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH134PICTUREFILESIZE586H  0޽h ? ̙33___PPT10.,+D' = @B Du' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Q%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(+! ( 0(    <: k  z$Algorithms for Logistic Regression %%$    v A?txp_fig" 4h SOURCEL\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ L(\bbeta) = \ell(\bbeta) + \log p(\bbeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1186PICTUREFILESIZE2303  6: Q  /where    v A9?txp_fig"At SOURCEX\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}= \arg\max_{\bbeta}\,L(\bbeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH936PICTUREFILESIZE2296   H: ?" WStandard algorithm: Newton-Raphson, a.k.a. iteratively reweighted least squares (IRLS)X   H: ?"bB j@IRLS is easily applied without any prior or with Gaussian prior.A l  ,N  ,N,$D 0t  ,6   ,6 ,$D 0   H: ?" ,6  vLIRLS not applicable with Laplacian prior: is not differentiable.Mv   v A1?txp_fig" U % < SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \lambda\|\bbeta\|_1 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH314PICTUREFILESIZE679t  :N  :N,$D 0  H: ?"   T*Alternative: bound optimization algorithms+  H`: ?"^` :N For ML logistic regression [Bhning & Lindsay, 1988], [Bhning,1992]. More general formulations [de Leeuw & Michailides, 1993], [Lange, Hunter, & Yang, 2000]. H  0޽h ? ̙33___PPT10n.@>+.JDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+" ~~@0y$ ~(  $ $ <: k z$Bound Optimization Algorithms (BOA) %%$ ~l   k$ ,$D 0B I$ B D8c?" z L$  v AC?txp_fig"@ SOURCE$\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}^{(2)} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH194PICTUREFILESIZE754~l   m$ ,$D  0B O$ B D8c?"$ $z Q$  v AF?txp_fig"@ SOURCE$\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}^{(3)} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH194PICTUREFILESIZE746l  u  n$ u ,$D 0r S$  v A@?txp_fig" 2 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \left. \right\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono,ORIGWIDTH 4,8754PICTUREFILESIZE239 T$ H: ?" m '  ASufficient (in fact more than sufficient) to prove monotonicity:BBB U$  0e0e    BC6DEFA jJ 8c8c     ?A)BCD|E||6w\W@  c"$`~  X$  v AB?txp_fig"  u  SOURCEl\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ L(\widehat{\bbeta}^{(t+1)}) \geq L(\widehat{\bbeta}^{(t)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH976PICTUREFILESIZE2435~ Z$  v AG?txp_fig"KTa$D  0  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \cdots \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH134PICTUREFILESIZE188 l  O o$ O,$D  0 [$ HD< ?"   YNotes:   ]$  v A<?txp_fig" Z SOURCE>\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ Q(\bbeta | \widehat{\bbeta}^{(t)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH486PICTUREFILESIZE1569 ^$ Hp: ?"  ashould be easy to maximize  _$ HD< ?" UO R EM is a BOA   l \  i$\ ,$D 0B B$ H D8c?"T T@ \  h$\ B "$ H DV?"B\ HB #$ N DV?"d 3$  60e0e    BC"DE|F4A 8c 8c8c     ?A)BCD|E||">`JPz{_6 (dd0qn&L_@zX@          c"$`* ^Lz M$  v AE?txp_fig"@ SOURCE$\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}^{(1)} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH194PICTUREFILESIZE676b a$  v AH?txp_fig"^w G ( SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ L(\bbeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH254PICTUREFILESIZE754 l 6   j$6  ,$D 0, H$  0e0e    BCDE4FA jJ 8c8c     ?A)BCD|E|| .4]|82a@ @    c"$`J   c$  v AI?txp_fig"6 , Z SOURCE>\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ Q(\bbeta | \widehat{\bbeta}^{(1)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH496PICTUREFILESIZE1511B d$ B D1?""F   l ^X / K l$X ^/ K,$D 08 N$  0e0e    BCDE4F jJ 8c8c     ?A)BCD|E|| .^JZa  0`W|j @    c"$`^X *  f$  v AJ?txp_fig"U / KZ SOURCE>\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ Q(\bbeta | \widehat{\bbeta}^{(2)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH496PICTUREFILESIZE1602B g$ B D1?"  &8  y$ $ Hl6< ?"& ?Optimization problem:  $  v A9?txp_fig"z-Jt SOURCEX\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}= \arg\max_{\bbeta}\,L(\bbeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH936PICTUREFILESIZE2296t  x$,$D 0 $  v A<?txp_fig"ZZJZ SOURCE>\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ Q(\bbeta | \widehat{\bbeta}^{(t)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH486PICTUREFILESIZE1569k@ ! w$!  $ H`K< ?"4g T O%where is such that & $  v A>?txp_fig"f !Qv SOURCEZ\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ Q(\bbeta | \widehat{\bbeta}^{(t)}) \leq L(\bbeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono.ORIGWIDTH 88,8756PICTUREFILESIZE2495 $ H\W< ?" ?  J ....with equality if and only if! 4$  v AD?txp_fig"  R SOURCE6\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bbeta = \widehat{\bbeta}^{(t)} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono.ORIGWIDTH 40,8754PICTUREFILESIZE973St  v$,$D 0  $ Hc< ?"y 4 GBound optimization algorithm: u$  v A?txp_fig" 6. SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}^{(t+1)} = \arg\max_{\bbeta}Q(\bbeta | \widehat{\bbeta}^{(t)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1356PICTUREFILESIZE9303H $ 0޽h ? ̙33:2___PPT10.+U{D' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*n$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*i$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*j$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*k$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*l$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*m$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Z$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*o$%(+L bFZFP+(E(  ( ( <y< k  oDeriving Bound Functions   (  v A<?txp_fig"> Z SOURCE>\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ Q(\bbeta | \widehat{\bbeta}^{(t)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH486PICTUREFILESIZE1569 ( H < ?"f  M#Many ways to obtain bound functions$* ( H < ?"M. zFor example, well known that Jensen s inequality underlies EM>-l c  +(c ,$D 0kt  ((,$D 0  ( H8< ?"P dVia Hessian bound: suppose is concave, with Hessian bounded below, e` (  v AH?txp_fig" d & SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ L(\bbeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH254PICTUREFILESIZE754n (  v AL?txp_fig"  4 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\cal H}(\bbeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH274PICTUREFILESIZE852 (  v AN?txp_fig"V SOURCE:\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\cal H}(\bbeta) \succeq - {\bf B}\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono.ORIGWIDTH 61,8756PICTUREFILESIZE1282 ( H< ?" S)where is a positive definite matrix.*Z (  v AO?txp_fig"    SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\bf B}\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH104PICTUREFILESIZE283t  3c  )( 3c ,$D 0F (  v AK?txp_fig" 3  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ L(\bbeta) \geq L(\bbeta') + (\bbeta-\bbeta')^T {\bf g}(\bbeta') - \frac{(\bbeta-\bbeta')^T {\bf B}(\bbeta-\bbeta')}{2} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2806PICTUREFILESIZE5704 !(  v AR?txp_fig") * c r SOURCEV\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \left.\nabla L(\bbeta)\right|_{\bbeta=\bbeta'} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH646PICTUREFILESIZE1666 "(  0e0e    BCBDEFA jJ 8c8c     ?A)BCD|E||l+0B s@  c"$`a b l  p *( p,$D 0 #( H(< ?"P p U+Can use r.h.s. as with , $(  v A<?txp_fig"H  fZ SOURCE>\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ Q(\bbeta | \widehat{\bbeta}^{(t)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH486PICTUREFILESIZE1569 '(  v AP?txp_fig" gV SOURCE:\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bbeta' = \widehat{\bbeta}^{(t)} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH456PICTUREFILESIZE1005H ( 0޽h ? ̙33___PPT10.+ND~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+(%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**(%(+ ` ,p(  , , << k&  w!Quasi-Newton Monotonic Algorithm ""!  , H< ?"v0 9Update equation ,  v AS?txp_fig"6. SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}^{(t+1)} = \arg\max_{\bbeta}Q(\bbeta | \widehat{\bbeta}^{(t)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1356PICTUREFILESIZE9127  , N< ?"B  Fis simple to solve, leads to , # | AW?txp_fig"yTL SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}^{(t+1)} = \widehat{\bbeta}^{(t)} + \Bmat^{-1}{\bf g}(\widehat{\bbeta}^{(t)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1336PICTUREFILESIZE7136l   , x ,$D 0) , N? ?"  ?This is a quasi-Newton algorithm, with B replacing the Hessian.$@'@ , N? 8c?" J  c-Unlike the Newton algorithm, it is monotonic.- H , 0޽h ? ̙33___PPT10n.W2&+ DB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(+7 11p400 1(  0 0 < ? kZ  {%Aplication to ML Logistic Regression &&%  0 H$? ?"H: :For logistic regression, can be shown that [Bhning,1992];,: 0 <? ?"7  ^Kroneker productR 0 # | A?txp_fig"sj SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\bpsi}{\mbox{\boldmath$\psi$}} \newcommand{\Deltamat}{\mbox{\boldmath$\Delta$}} \newcommand{\xv}{{\bf x}} \newcommand{\pv}{{\bf p}} \pagestyle{empty} \begin{document} \[ {\cal H}(\bbeta) \succeq - \frac{1}{2} [{\bf I} - {\bf 1} {\bf 1}^T /m] \otimes \sum_{j=1}^n \bpsi(\xv_j) \bpsi(\xv_j)^T \equiv -\Bmat\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2598PICTUREFILESIZE 14451V !0  0e0e    BGCDE(FA A8c 8c8c     ?A)BCD|E|| Gfvj5mS   G @   S"R{ l lT  30lT ,$D 0 "0 N"? 8c?"l kCAlso easy to compute the gradient and finally plug into  D @ $0 # | A8c?txp_fig" ; ~v SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\bpsi}{\mbox{\boldmath$\psi$}} \newcommand{\xv}{{\bf x}} \newcommand{\yv}{{\bf y}} \newcommand{\pv}{{\bf p}} \pagestyle{empty} \begin{document} \[ {\bf g}(\bbeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono.ORIGWIDTH 22.8754PICTUREFILESIZE829 &0 # | AW?txp_fig" T TL SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}^{(t+1)} = \widehat{\bbeta}^{(t)} + \Bmat^{-1}{\bf g}(\widehat{\bbeta}^{(t)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1336PICTUREFILESIZE7136Xl  /V 40 /V,$D 0 *0 B8? ?"   S!Under a ridge-type Gaussian prior!d ,0 # | Ae?txp_fig" *' SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}^{(t+1)} = (\Bmat+\lambda{\bf I})^{-1} \left({\bf B}\widehat{\bbeta}^{(t)} + {\bf g}(\widehat{\bbeta}^{(t)})\right) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1938PICTUREFILESIZE 10424 -0 BE? ?"~ 6/V Kcan be computed off-line. 00 # | A"8c?txp_fig"  . ~ SOURCEb\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \log p(\bbeta) \propto -(\lambda/2)\, \|\bbeta\|_2^2 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1206PICTUREFILESIZE6790d 10  0e0e    BGCDE(F 8c 8c8c     ?A)BCD|E|| Gfvj5mS   G @   S"I-{ zB 20 < D8c?")> >H 0 0޽h ? ̙33___PPT10+qX-D~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*30%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*40%(+E V=N==8<(  8 8 <U? k ~(Aplication to LASSO Logistic Regression ))(  8  v A#?txp_fig"m`X SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ L(\bbeta) = \ell(\bbeta) - \lambda \, \|\bbeta\|_1 = \ell(\bbeta) - \lambda \sum_{l=1}^k |\beta_k| \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2106PICTUREFILESIZE4667 8 <c? ?" OFor LASSO logistic regressionwl  98,$D 0 8 <h? ?" S!already bounded via Hessian bound! 8 # 0e0e    BCDEF jJ 8c8c     ?A)BCD|E|||u-N@  c"$`  `ql v :8v,$D 0  8 <l? ?"va M...need bound for log prior  8 # 0e0e    BCDEF jJ 8c8c     ?A)BCD|E|||u-N@  c"$`N,l 1` =81`,$D 0Ht -  ` <8-  `,$D 0 8 H Ab ??"  B $8 H D8c?"-  B *8 B D8c?"!  ^ ,8  v Ag?txp_fig"7 $ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ -|\beta| \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH224PICTUREFILESIZE451 /8 <dx? ?"]y ` ^quadratic bound Z 18  v Aj?txp_fig"- k   SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \beta' \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH104PICTUREFILESIZE430\ 38  v Ak?txp_fig"5 ?$ " SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ -\beta' \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH194PICTUREFILESIZE467t 1@  ;81@ ,$D 0  8 <? ?"r  MEasy to show that, for any j 8  v Ai?txp_fig"  ( SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \beta'\!, \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono.ORIGWIDTH 10,8754PICTUREFILESIZE463 48 <? ?"g R@  H...with equality iff l 68  v Am?txp_fig"r9 : 2 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \beta = \beta' \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH344PICTUREFILESIZE659 88  v Al?txp_fig" 1= 0( SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ -|\beta| \geq - \frac{1}{2}\left( \frac{\beta^2}{|\beta'|} + |\beta'|\right) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1246PICTUREFILESIZE2992H 8 0޽h ? ̙33___PPT10+\KD' = @B Du' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*98%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*:8%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*=8%(+4 4y4 $<0 3(  < < <? k ~(Aplication to LASSO Logistic Regression ))(  < <? ?"3 PAfter dropping additive terms,  < <? ?"z 7where  <  v Ah?txp_fig"`M d\ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ \vv^{(t)} = \Bmat\widehat{\bbeta}^{(t)} + {\bf g}(\widehat{\bbeta}^{(t)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1096PICTUREFILESIZE6022 <  v Aq?txp_fig"'4_ SOURCEp\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ \Lambdamat^{(t)} = \mbox{diag}\left\{ |\hat{\beta}_1^{(t)}|^{-1},\dots,|\hat{\beta}_k^{(t)}|^{-1} \right\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1716PICTUREFILESIZE8248 <  v A-?txp_fig"qD\  SOURCEh\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ Q(\bbeta|\widehat{\bbeta}^{(t)}) = 2\bbeta^T \vv^{(t)} -\bbeta^T (\Bmat+\lambda\Lambdamat^{(t)})\bbeta \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2008PICTUREFILESIZE 10852 < B? ?"$ D  PThe update equation is an IRRR < # | Ao?txp_fig" ZR D SOURCE(\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}^{(t+1)} = (\Bmat+\lambda\Lambdamat^{(t)})^{-1}\vv^{(t)} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1336PICTUREFILESIZE6102 < B? ?"e   Kwhich can be rewritten as < # | An?txp_fig"  R SOURCE6\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\Upsilonmat}{\mbox{\boldmath$\Upsilon$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}^{(t+1)} = \Upsilonmat^{(t)} \left( \Upsilonmat^{(t)} \Bmat\Upsilonmat^{(t)} + \lambda{\bf I}\right)^{-1}\Upsilonmat^{(t)} \vv^{(t)}, \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2136PICTUREFILESIZE9682 < B? ?"!aA 7where  !< # | A?txp_fig"*9 KPH SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\Upsilonmat}{\mbox{\boldmath$\Upsilon$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ \Upsilonmat^{(t)} = \{|\hat{\beta}_1^{(t)}|^{1/2},\dots ,|\hat{\beta}_k^{(t)}|^{1/2}\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1486PICTUREFILESIZE7448H < 0޽h ? ̙33; 44@=4(  @ @  v An?txp_fig"yR SOURCE6\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\Upsilonmat}{\mbox{\boldmath$\Upsilon$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}^{(t+1)} = \Upsilonmat^{(t)} \left( \Upsilonmat^{(t)} \Bmat\Upsilonmat^{(t)} + \lambda{\bf I}\right)^{-1}\Upsilonmat^{(t)} \vv^{(t)}, \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2136PICTUREFILESIZE9682 @ <G k ~(Aplication to LASSO Logistic Regression ))(  @ <LG ?"^~ EThe update equation @ <G ?" Ihas computational cost 7 l 0  @0 ,$D 0  @ <G ?" 50  sAMay not be OK for kernel classification for largeA @ B|!G ?"; vDOK for linear classification if not too large. Dn  @ # | Au?txp_fig"z  * SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\Upsilonmat}{\mbox{\boldmath$\Upsilon$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ (k=d) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH366PICTUREFILESIZE1677b  @ # | Av?txp_fig"1  x" SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\Upsilonmat}{\mbox{\boldmath$\Upsilon$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ d \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE540n @ # | Ax?txp_fig"D C - * SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\Upsilonmat}{\mbox{\boldmath$\Upsilon$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ (k=n) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH376PICTUREFILESIZE1661b @ # | Ay?txp_fig"  " SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\Upsilonmat}{\mbox{\boldmath$\Upsilon$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ n \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE498@ @ <PIG ?" X ,$ 0 `This is the cost of standard IRLS for ML logistic regression ...but now with a Laplacian prior.`h @  v Az?txp_fig"{ 2 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\Upsilonmat}{\mbox{\boldmath$\Upsilon$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ O((km)^3) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH486PICTUREFILESIZE3069H @ 0޽h ? ̙33___PPT10+VD' = @B DF' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(+8+0+@G  +H DD$DZD(  D D <ZG k :Sequential Update Algorithm for LASSO Logistic Regression ;;:  D <\aG ?"^~ = Recall that  D  v Ar?txp_fig"g | SOURCE`\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ L(\bbeta) = \ell(\bbeta) - \lambda \, \|\bbeta\|_1 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1116PICTUREFILESIZE2051& D <xmG ?" zLet s bound only via the Hessian bound, leaving =^ D  v A{?txp_fig"z-, SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \ell(\bbeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH224PICTUREFILESIZE806~  D  v A|?txp_fig"<L SOURCE0\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ - \lambda \, \|\bbeta\|_1 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH424PICTUREFILESIZE700%l ( @7 $D( @7,$D 0  D <dG ?"( H  `Maximizing only w.r.t. the  th component of 0\ D  v A?txp_fig"H @6  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[\bbeta \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono,ORIGWIDTH 7,8754PICTUREFILESIZE395H D  v A}?txp_fig"l J   SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[l\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH44PICTUREFILESIZE250 D  v A?txp_fig" $ V SOURCE:\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ \widehat{\beta}_l^{(t+1)} = \arg\max_{\beta_l} Q(\bbeta|\widehat{\bbeta}^{(t)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1346PICTUREFILESIZE8973^ D  v A?txp_fig" 7  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ \widehat{\beta}_s^{(t+1)} = \widehat{\beta}_s^{(t)} , \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH646PICTUREFILESIZE3792 D <DG ?"  4 5for  D  v A?txp_fig" F v .F> SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ s\neq l \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH246PICTUREFILESIZE1178 D BhG ?" "  "D # | A?txp_fig"t SOURCEl\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ Q(\bbeta|\widehat{\bbeta}^{(t)}) = 2\, \bbeta^T \vv^{(t)} -\bbeta^T \Bmat\bbeta - 2\,\lambda\|\bbeta\|_1 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2058PICTUREFILESIZE 10407H D 0޽h ? ̙33~___PPT10^+BDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$D%(+rK BB0H[B(  H H  v A?txp_fig"V SOURCE:\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ \widehat{\beta}_l^{(t+1)} = \arg\max_{\beta_l} Q(\bbeta|\widehat{\bbeta}^{(t)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1346PICTUREFILESIZE8973 H <$G k :Sequential Update Algorithm for LASSO Logistic Regression ;;:  H <G ?"?_ FThe update equation l H / 5  /H/ H 5 ,$D 0  H <G ?"H / 5  e3Can be shown that updating all components has cost 3j #H  v A?txp_fig" C . , SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\Upsilonmat}{\mbox{\boldmath$\Upsilon$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ O(nkm) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH416PICTUREFILESIZE2635l _ q   0Hq _  ,$D 0 $H <G ?"_ q   Hmay be much less than p &H  v Az?txp_fig" ) 2 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\Upsilonmat}{\mbox{\boldmath$\Upsilon$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ O((km)^3) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH486PICTUREFILESIZE3069 'H <G ?" e ",$ 0 Usually also uses fewer iterations, since we do not bound the prior and is  incremental Y%l   -H ,$ 0 H <G ?"5 V$has a simple closed-form expression $@ s  ,Hs B  H H DV?"  B H N DV?"0  0 B H B D8c?"'>  . B H B D8c?"s+ cB H B D8c?"]/ %/ R H  v A?txp_fig"c u  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ z \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH64PICTUREFILESIZE495z H  v A?txp_fig"; < SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \mbox{soft}(z,\tau) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH456PICTUREFILESIZE2360X H  v A?txp_fig"b t  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \tau \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH64PICTUREFILESIZE309\ H  v A?txp_fig"    SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ -\tau \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH154PICTUREFILESIZE359~ +H  v A?txp_fig"qc < SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\beta}_s^{(t+1)} = \mbox{soft}\left(\widehat{\beta}_s^{(t)} + \frac{g_s(\widehat{\bbeta}^{(t)})}{B_{ss}}, \frac{\lambda}{B_{ss}}\right) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1788PICTUREFILESIZE 13204H H 0޽h ? ̙33C;___PPT10+RĐD' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/H%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0H%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'H%(+8+0+'HX  +l$ B:(    <P*X kp :Sequential Update Algorithm for Ridge Logistic Regression ;;:    v A?txp_fig"V SOURCE:\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\Lambdamat}{\mbox{\boldmath$\Lambda$}} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \newcommand{\vv}{{\bf v}} \pagestyle{empty} \begin{document} \[ \widehat{\beta}_l^{(t+1)} = \arg\max_{\beta_l} Q(\bbeta|\widehat{\bbeta}^{(t)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1346PICTUREFILESIZE8973  <9X ?"?_ ]+With a Gaussian prior, the update equation +  <=X ?"{ \*also has a simple closed-form expression v*@   v A?txp_fig"  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \hat{\beta}_s^{(t+1)} = \frac{1}{B_{ss}+\lambda}\left(B_{ss}\,\hat{\beta}_s^{(t)} + g_s(\widehat{\bbeta}^{(t)})\right) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1888PICTUREFILESIZE 11797l c )  c ) ,$D 0  <JX ?"c )  XDFor , we get the update rule for ML logistic regression. h   v A?txp_fig"7 -T , SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \lambda = 0 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH284PICTUREFILESIZE959-z     ,$D 0B  B D8c?"    BpWX ?"t z  sImportant issue: how to choose the order of update, i.e., which component to update next?  B[X ?"x  R>In all the results presented, we use a simple cyclic schedule."  B_X ?" We re currently investigating other alternatives (good, but cheap).H  0޽h ? ̙33___PPT10.G/08+7D~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+ zrL (  L L <fX k c Related Work   L HjX ?" zF MSequential update for the relevance vector machine (RVM) [Tipping & Faul, 2003] Comments: the objective function of the RVM is not concave, results may depend critically on initialization and order of update. *P {,PlxB  L B D?"nl  L,$D 0  L HsX ?"l 5Kernel logistic regression; the import vector machine (IVM) [Zhu & Hastie, 2001] Comments: sparseness not encouraged in the objective function (Gaussian prior) but by early stopping a greedy algorithm. ,-Q rCB  L B D?"`2 l L m  LL m ,$D 0B  L B D?" C   L H|X ?"VL m  BEfficient algorithm for SVM with penalty [Zhu, Rosset, Hastie, & Tibshirani, 2003] Comments: efficient, though not simple, algorithm; the SVM objective is different from the logistic regression objective.,-Z r1'}V L  v A?txp_fig"/ X 0  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ l_1 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE421l   L ,$D 0 L HX ?"V  Least angle regression (LAR) [Efron, Hastie, Johnstone, & Tibshirani, 2002] Comments: as far as we know, not yet/easily applied to logistic regression.,-L C,NB L B D?" e H L 0޽h ? ̙33___PPT10+v?OD' = @B Du' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(+:# F>#\(  \ \ < A?"Nl ,$@ 0 \ <X k jExperimental Results _ \ HX ?"a@ Three standard benchmark datasets: Crabs, Iris, and Forensic Glass. Three well-known gene expression datasets: AML/ALL, Colon, and Yeast. l    \ ,$D 0  \ HPX ?"  1Penalty weight adjusted by cross-validation.221^ \  v A?txp_fig" _$ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \lambda \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE523L \ HwX ?",$ 0 DComparison with state-of-the-art classifiers: RVM and SVM (SVMlight)EED \ HPX ?"JD,$ 0 bSummary of datasets:l o  "\ o,$D 0  \ H|X ?"o" |.Not exactly CV, but 30 different 50/12 splits //.B \ B D8c?". .:&l @ , !\ @,,$D 0B \ B D8c?"X X ]" \  0e0e    BCXDE(FA 8c 8c8c     ?A)BCD|E|| '''r'&'X8'{ @   c"$`  \ HX ?"@E , jNo CV, fixed standard split H \ 0޽h ? ̙33___PPT10t.Lk+D ' = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* \%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!\%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"\%(+p+0+\X  ++0+\X  +#   `D (  ` ` <X k jExperimental Results K ` HX ?"IU wAll kernel classifiers (RBF Gaussian and linear). For RBF, width tuned by CV with SVM, and used for all other methods.xxw  ` HX ?"E G ,$ 0 Linear classification of AML/ALL (no kernel): 1 error, 81 (of 7129) features (genes) selected. V.O x ` <dX ?"T,$ 0 B[Krishnapuram, Carin, Hartemink, and Figueiredo, 2004 (submitted)]CCP   {l mA,  `Am, ,$D 0z ` < A?"m@~  ` HX ?" #  bNumber of errors  ` HX ?"    aNumber of kernels B  ` B D8c?"z  B  ` B D8c?"z    ` B jJ?#" `  ` HX ?"A.; VResults:  ` HtX ?"t  >0 ` B|X V?#" ` ,  rBMSLR Bayesian multinomial sparse logistic regression BMGLR Bayesian multinomial Gaussian logistic regressionssH ` 0޽h ? ̙33___PPT10.Pp m+SD' = @B DF' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*`%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* `%(+8+0+ `X  +#  d(  d d <^ ki  rNon-linear Feature Selection # d <l^ ?"d 6 wWe have considered a fixed dictionary of functions i.e., feature selection is done on parameters appearing linearly w  d  v A$?txp_fig" H@ SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ f({\bf x},\mbox{\boldmath$\beta$}) = \mbox{\boldmath$\beta$}^T \mbox{\boldmath$\psi$}({\bf x}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH796PICTUREFILESIZE1662  d  v A?txp_fig"b _  SOURCEf\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\bpsi}{\mbox{\boldmath$\psi$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ P \left( y=1|\xv{},\bbeta \right) = \frac{\exp \left( \bbeta^T \bpsi({\bf x}) \right)}{1 + \exp \left( \bbeta^T \bpsi({\bf x}) \right)} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1816PICTUREFILESIZE5797  d < ^ ?"Dd  f4or  generalized linearly   d  v AT?txp_fig"7f^ SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \newcommand{\bpsi}{\mbox{\boldmath$\psi$}} \pagestyle{empty} \begin{document} \[ \bpsi({\bf x}) = [ \psi_1 ({\bf x}),\dots , \psi_k ({\bf x})]^T, \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1216PICTUREFILESIZE5392xl h E? dh E?,$D 0H d H8*^ ?"h E  Let us now look at  non-linear parameters , i.e., inside the dictionary:I d  v AZ?txp_fig" ?l SOURCEP\documentclass{article} \usepackage{amsmath,amssymb} \newcommand{\bpsi}{\mbox{\boldmath$\psi$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \pagestyle{empty} \begin{document} \[ \bpsi({\bf x},\btheta) = [ \psi_1 ({\bf x},\btheta),\dots , \psi_k ({\bf x},\btheta)]^T \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1506PICTUREFILESIZE7134H d 0޽h ? ̙33___PPT10n.W+w&+3DB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(++ %%  !hK%(  h h <7^ ki  rNon-linear Feature Selection  h H4>^ ?"J& CFor logistic regression, P h  v A?txp_fig"D SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\bpsi}{\mbox{\boldmath$\psi$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ P \left( y=1|\xv{},\bbeta,\btheta \right) = \frac{\exp \left( \bbeta^T \bpsi({\bf x},\btheta) \right)}{1 + \exp \left( \bbeta^T \bpsi({\bf x},\btheta) \right)} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono0ORIGWIDTH205,8756PICTUREFILESIZE6755 l Q   hQ ,$D 0  h HHK^ ?"  VWe need to further constraint the problem. ...consider parameterizations of the type: W h  v A?txp_fig"Q ZR SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\bpsi}{\mbox{\boldmath$\psi$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bpsi({\bf x},\btheta) = \bpsi(\sqrt{\mbox{diag}(\btheta)}\, {\bf x}) = \bpsi([\sqrt{\theta_1} x_1,\dots,\sqrt{\theta_d} x_d]^T) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono0ORIGWIDTH271,8756PICTUREFILESIZE5545Tl   !h ,$D 0 h 6Z^  7  4 Polynomial b h  v A?txp_fig"x    SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ K({\bf x},{\bf x}',\btheta) = \left(1+\sum_{l=1}^d \theta_l\, x_l\, x'_l \right)^r \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1658PICTUREFILESIZE 10864p h  v A?txp_fig"{` . SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ K({\bf x},{\bf x}',\btheta) = \exp\left( - \sum_{l=1}^d \theta_l\; ( x_l - x'_l)^2 \right) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1988PICTUREFILESIZE 12680 h 6o^   2Gaussian  h 6s^    6 For kernels: H h 0޽h ? ̙33___PPT10.X .-+D~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* h%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!h%(+1 &--0p,(  p p H{^ ?"lG lBThis corresponds to a different scaling of each original feature. C8   p  p Hp^ ?"  Z0Sparseness of feature selection. 1 p  v Ac?txp_fig" TL SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\bpsi}{\mbox{\boldmath$\psi$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \btheta\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE325`B p 0D8c p H^ ?"b  Q'We can also adopt Laplacian prior for (  p  v Ac?txp_fig" < QTL SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\bpsi}{\mbox{\boldmath$\psi$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \btheta\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE325  p  v A?txp_fig"9ZR SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\bpsi}{\mbox{\boldmath$\psi$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bpsi({\bf x},\btheta) = \bpsi(\sqrt{\mbox{diag}(\btheta)}\, {\bf x}) = \bpsi([\sqrt{\theta_1} x_1,\dots,\sqrt{\theta_d} x_d]^T) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono0ORIGWIDTH271,8756PICTUREFILESIZE5545  p <ܠ^ kN  oFeature Scaling/Selection z o p o,$D 0  p # | A?txp_fig"  6 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \left(\widehat{\bbeta},\widehat{\btheta}\right)= \arg\max_{\bbeta,\btheta}\, L(\bbeta,\btheta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono0ORIGWIDTH130,8756PICTUREFILESIZE3877 p N^ ?"  ?Estimation criterion: p N^ ?":  ALogistic log-likelihoodB p N D8c?"\ b t p # | A?txp_fig" o 0 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ L(\bbeta,\btheta) = \ell(\bbeta,\btheta) - \lambda_1 \|\bbeta\|_1 - \lambda_2 \|\btheta\|_1 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1896PICTUREFILESIZE3398H p 0޽h ? ̙33___PPT10n.\ĵ+7|cDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(+M *E"E@>l D(  l l <^ k] *EM Algorithm for Feature Scaling/Selection++*  !l H^ ?"t9 ?Optimization problem:@l X =lX,$D 0 $l N^ ?"Xyx \2We use again a bound optimization algorithm (BOA),3 &l # | A?txp_fig"5<4 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \left( \widehat{\bbeta}^{(t+1)},\widehat{\btheta}^{(t+1)}\right) = \arg\max_{\bbeta,\btheta}Q(\bbeta,\btheta | \widehat{\bbeta}^{(t)},\widehat{\btheta}^{(t)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2158PICTUREFILESIZE 14498 'l N^ ?"i j@with Hessian bound for and the quadratic bound for A )l # | AA?txp_fig"E s $ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \ell(\bbeta,\btheta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono.ORIGWIDTH 34,8756PICTUREFILESIZE1074 +l # | A?txp_fig"B SOURCEn\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \|\cdot\|_1 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono.ORIGWIDTH 22,8754PICTUREFILESIZE243 l  s  >l s ,$D 0 ,l N^ ?"8 gX  6 Maximizing  > /l # | A?txp_fig"P p O |t SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ Q(\bbeta,\btheta | \widehat{\bbeta}^{(t)},\widehat{\btheta}^{(t)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH826PICTUREFILESIZE6017 0l N^ ?" 8 sX  j@can t be done in closed-form. ! 1l Nf ?"u   X.Easy to maximize w.r.t. , with fixed. / 4l # | A?txp_fig"| { ~ SOURCEb\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bbeta\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE728 6l # | A?txp_fig" t ]  SOURCEd\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \btheta\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE558 7l Nf ?"   _Maximization w.r.t. done by conjugate gradient; necessary gradients are easy to derive. ` 8l # | A?txp_fig"c   SOURCEd\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \btheta\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE558 9l  v AM?txp_fig"`0-D< SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \left(\widehat{\bbeta},\widehat{\btheta}\right)= \arg\max_{\bbeta,\btheta}\left\{ \ell(\bbeta,\btheta) - \lambda_1 \|\bbeta\|_1 - \lambda_2 \|\btheta\|_1\right\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2388PICTUREFILESIZE 13503` +B#style.visibility<*=l%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*>l%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x Bj ?"Q  l94.4 @` =x BLj ?" {SVM (linear kernel) @`  Accuracy (%) ~ x H o?"   x H_j ?"e c[Ben-Dor et al, 2000]xB x B D8c?"0xB x B D8c?"))xB x@ B D8c?"m( x Hcj ?"r [Krishnapuram et al, 2002]8  xB x@ B D8c?" xB x B D8c?"n  x Hhj ?"X   6Tipically around 25~30 genes selected, i.e., non-zero 776 x  v AU?txp_fig" B  SOURCEf\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \theta_i\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE701H x 0޽h ? ̙33y___PPT10Y+D=' = @B +    ` |= (  |r | < A?"3 5 | <9j ar /Top 12 Genes for AML/ALL (sorted by mean |qi| )\0*  /  | Hȇj ?"~U)  K*  | Hj ?" d 1  K*  | Hj ?"  K*  | Hdj ?"E Q% e  K*  | H@j ?"  K*  | Hj ?" 0/ :Agree with [Golub et al, 1999]; many others in the top 25.;;;x | < ?#" `5ux | < ?#" `-( | H|j ?"x+r }7Antibodies to MPO are used in clinical diagnosis of AML888H | 0޽h ? ̙33y___PPT10Y+D=' = @B +Z qip (  r  < A?"rP   <j jr +Top 12 Genes for Colon (sorted by mean qi )B,' +   Hj ?" K*  Hj ?"Mdm K*   Hزj ?"9 K*   HHj ?" K*   Hj ?"9 DY K*   H|j ?": _ 4 m'Known to be implicated in colon cancer.(((H  0޽h ? ̙33___PPT10i.Z/pS +D=' = @B + B(     Hj ?"Z(  2PART II Non-trivial Bounds for Sparse Classifiers332H  0޽h ? ̙33___PPT10i.pş+D=' = @B +^3 4-,-: ,(    <Hj [E{ T Introduction     # | A?txp_fig"Y2* SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ {\cal D} = \left\{ ({\bf x}_1,y_1), ({\bf x}_2,y_2), \dots, ({\bf x}_n,y_n)\right\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1566PICTUREFILESIZE2557   Bj f! n8Training data (here, we consider only binary problems):8    <Lj 9 j4assumed to be i.i.d. from an underlying distribution4 R    v A?txp_fig"!T7  SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ p({\bf x},y) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH284PICTUREFILESIZE795l w% I $e w ,$D 0  Hj ?#" `3  P&Key question: how are the two related?'   B 8c?#" `w% Il p :p,$D 0qt ( *G,$D 0  N8j ?"(  =Given a classifier  (  v A?txp_fig"XP SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \pagestyle{empty} \begin{document} \[ f(\cdot,\bbeta): \mathbb{R}^d\rightarrow \{-1,+1\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1226PICTUREFILESIZE4656$ t p  8p ,$D 0  N|j ?"p  rHTrue generalization error (not computable, is unknown):I`  # | A?txp_fig"9   SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ p({\bf x},y) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH284PICTUREFILESIZE795* 4  v A?txp_fig"'o  nf SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \pagestyle{empty} \begin{document} \[ E_{\mbox{\tiny true}}(f) = E_{p({\bf x},y)}[y\neq f({\bf x})] \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1376PICTUREFILESIZE6817qt &  9& ,$D 0  N@n ?"& F  7 Sample error: 7 # | A8c?txp_fig#"   RJ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ E_{\mbox{\tiny sample}}(f,{\cal D}) = \frac{1}{n}\sum_{i=1}^n {\bf 1}_{y_i\neq f({\bf x}_i)} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1486PICTUREFILESIZE9008H  0޽h ? ̙33___PPT10.|+D~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(+, n&f&%(    <\n [ { ^PAC Performance Bounds    Htn ?"2 f<PAC (probably approximately correct) bounds are of the form:=   H\n ?"2z R Eand hold independently of     v A?txp_fig"mnRJ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ Pr\{{\cal D}:\; E_{\mbox{\small true}}(f) \geq \mbox{bound}(f,{\cal D},\delta)\}\leq \delta, \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2086PICTUREFILESIZE9511\   v A ?txp_fig"* h" SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ p({\bf x},y). \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono.ORIGWIDTH 31,8754PICTUREFILESIZE852dl JA  JA ,$D 0  H0n ?"J j  J Usually, bounds have the form: !   v A?txp_fig"1 A LD SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \mbox{bound}(f,{\cal D},\delta) = E_{\mbox{\tiny sample}}(f) + \Delta(f,{\cal D},\delta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2016PICTUREFILESIZE9284Ol  H H H H,$D 0B  N D8c??"(H    # | A8c?txp_fig#"  2 c 0( SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \Delta(f,{\cal D},\delta) \stackrel{n\rightarrow\infty}{\longrightarrow}0, \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH886PICTUREFILESIZE4793  NdDn 8c?#"   "&B 9uniformly over d   v A ?txp_fig"3 H" SOURCE\documentclass{article} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \[ p({\bf x},y). \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono.ORIGWIDTH 31,8754PICTUREFILESIZE852H  0޽h ? ̙33___PPT10.v@+|D~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+. { (    <Qn [ { ^PAC Performance Bounds    NWn 8c?#"  i  L"There are several ways to derive #  # | A8c?txp_fig#"  ' =D SOURCE(\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \Delta(f,{\cal D},\delta)\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH506PICTUREFILESIZE2900l 2 2,$D 0-  N bn 8c?#"  2R =- Vapnik-Chervonenkis (VC) theory (see, e.g., [Vapnik, 1998])$>.>3   Nhn 8c?#"  }D,$ 0 e;VC usually leads to trivial bounds (>1, unless n is huge).<Ml +lr  +lr ,$D 0)   Ncn 8c?#"  +aK A- Compression arguments [Graepel, Herbrich, & Shawe-Taylor, 2000]B*B   Nqn 8c?#"  hllr ,$ 0 Compression bounds are not applicable to point sparse classifiers of the type herein presented, or of the RVM type [Herbrich, 2000].sT   Nvn 8c?#"   m ,$ 0 @We apply PAC-Bayesian bounds [McAllester, 1999], [Seeger, 2002].A$AH  0޽h ? ̙33SK___PPT10+.}+pD' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(+8+0+ n  +k cc8c(    <܂n [{ XSome Definitions  J  Hln ?":EZ,$ 0 f : some point  estimate of . Let 4   v A?txp_fig"`XP$ 06 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE925    v A?txp_fig"8 @$ 0" SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bbeta \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE728Zl  2,$ 0  H$n ?" e  Ba Laplacian centered at    v A?txp_fig" L SOURCE0\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\boldeta}{\mbox{\boldmath$\eta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ q(\bbeta|\widehat{\bbeta},\boldeta) \propto \exp\left\{ -\sum_l \eta_l |\beta_l - \hat{\beta}_l| \right\}, \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1888PICTUREFILESIZE 12532p   v A?txp_fig"@6 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE925f  HPn ?",$ 0 we ll call this the  posterior , although not in the usual sense.B )  v A ?txp_fig"CYr($ 0x SOURCE\\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ f({\bf x},\bbeta) = \mbox{sign}({\bf x}^T \beta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1026PICTUREFILESIZE5596l " @  3" @ ,$D 0  H4n ?" DPoint classifier (PC) at   H n ?"N"(  `6the one we re interested in  B V?#" `(0 @ ^   v A?txp_fig"H " SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bf x: \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH134PICTUREFILESIZE600 *  v A ?txp_fig" \ SOURCE@\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \hat{y} = {\bf x}^T\widehat{\bbeta} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH446PICTUREFILESIZE2414l  -  4 - ,$D 0^   v A?txp_fig" x " SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bf x: \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH134PICTUREFILESIZE600  HLn ?"   DGibbs classifier (GC) at b   v A ?txp_fig" . ) & SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bbeta_s \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH134PICTUREFILESIZE971   Hn ?"  "  8a sample from  "  v A ?txp_fig" - <4 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\boldeta}{\mbox{\boldmath$\eta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ q(\bbeta|\widehat{\bbeta},\boldeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH476PICTUREFILESIZE3037 ,  v A?txp_fig"  N SOURCE2\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \hat{y} = {\bf x}^T \bbeta_s \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH486PICTUREFILESIZE2449xl &B X 8B &X,$D 0d # # | A?txp_fig"h 8 " SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bf x: \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH134PICTUREFILESIZE600 $ Nv ?"&B b  L"Bayes voting classifier (BVC) at # 7 # | A8c?txp_fig"h X` SOURCED\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\boldeta}{\mbox{\boldmath$\eta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \hat{y} = \mbox{sign}\left[\int q(\bbeta|\widehat{\bbeta},\boldeta)\, \mbox{sign}({\bf x}^T \bbeta)\, d\bbeta \right] \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1868PICTUREFILESIZE 11159H  0޽h ? ̙33___PPT10.p.K'+Wa-D' = @B Du' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(+6 **/*(    <Dv [i{ R Key Lemmas   8  w  Hdv ?" Lemma 1: for any , the decision of the PC with is the same as that of a BVC based on any symmetric posterior centered on &^    v A?txp_fig"$ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\bf x} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE496p    v A?txp_fig"`6 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE925    v A ?txp_fig"80 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\boldeta}{\mbox{\boldmath$\eta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ q(\bbeta|\widehat{\bbeta},\boldeta)\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH476PICTUREFILESIZE3037t   v A?txp_fig"j8 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}. \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH114PICTUREFILESIZE985G  HAv ?",$ 0 ?Proof: a simple pairing argument (see, e.g., [Herbrich, 2002]).@:@L l BH  BH ,$D 0  HGv ?"BH  FLemma 2: for any  posterior , the generalization error of the BVC is less than twice that of the GC. &x mx   v A ?txp_fig"R c0 ] 80 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\boldeta}{\mbox{\boldmath$\eta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ q(\bbeta|\widehat{\bbeta},\boldeta)\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH476PICTUREFILESIZE3037$  HUv ?" ,$ 0 jProof: see [Herbrich, 2002].E  H[v ?" x2,$ 0 =Conclusion: we can use PAC-Bayesian bounds for GC for our PC.> 3>H  0޽h ? ̙33  ___PPT10 .g\+dD ' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(++0+v  ++0+v  ++0+v  +d7 ..I.(    <@jv [{ \PAC-Bayesian Theorem    Npv 8c?#"  j h>Let be our prior (meaning it is independent of )?j   # | A8c?txp_fig#"  `p( SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ p(\bbeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH236PICTUREFILESIZE1711f   # | A8c?txp_fig#"  xX& SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\cal D} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH104PICTUREFILESIZE668   Nhv 8c?#"   h>Let be our posterior (meaning it may depend on )?f  # | A8c?txp_fig#"  FZ& SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\cal D} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH104PICTUREFILESIZE668j  # | A8c?txp_fig#"  $X( SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ q(\bbeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH236PICTUREFILESIZE1637l T+8 T+8,$D 0  NLv 8c?#"  T t V,Generalization error for a Gibbs classifier:-t  # | A8c?txp_fig#"  +8( SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \pagestyle{empty} \begin{document} \[ E_{\mbox{\tiny gen}}(q) = E_{q(\beta)}\left[E_{p({\bf x},y)}[y\neq f({\bf x},\bbeta)]\right] \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1836PICTUREFILESIZE9633l 0   0  ,$D 0  Nv 8c?#"  0 P  a7Expected sample/empirical error for a Gibbs classifier:8  # | A8c?txp_fig#"  U  R SOURCE6\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \pagestyle{empty} \begin{document} \[ E_{\mbox{\tiny emp}}(q) = E_{q(\beta)}\left[\frac{1}{n}\sum_{i=1}^n {\bf 1}_{y_i\neq f({\bf x}_i,\bbeta)}\right] \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1738PICTUREFILESIZE 11057f  Nv 8c?#"  :Z,$ 0 vMcAllester s PAC-Bayesian theorem relates these two errors.<H  0޽h ? ̙33SK___PPT10+.+I D' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+v  +l6 B0:0'/(    <v [{ \PAC-Bayesian Theorem    Nv 8c?#"  `\]L $Theorem: with as defined above, the following inequality holds with probability at least over random training samples of size   # | A8c?txp_fig#"  |r.& SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ p(\bbeta),\; q(\bbeta),\;E_{\mbox{\tiny gen}}(q),\;E_{\mbox{\tiny emp}}(q) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1446PICTUREFILESIZE7996l  # | A8c?txp_fig#"  }Se* SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ (1-\delta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH346PICTUREFILESIZE1340V  # | A8c?txp_fig#"  = SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ n \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE514, l DTz  &TDz ,$D 0  NPv 8c?#"  DTt  0where   Nv 8c?#"  HZ Kz  d:is the Kullback-Leibler div. between two Bernoullies, and ;  # | A8c?txp_fig#"  `  SOURCEp\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ D_{\mbox{\tiny Ber}}(a,b) = a\ln(a/b)+(1-a)\ln[(1-a)/(1-b)] \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2506PICTUREFILESIZE9959l R  ' R,$D 0 " # | A"8c?txp_fig#"    @8 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ D_{\mbox{\tiny KL}}(q||p) = \int q(\bbeta)\ln\frac{q(\bbeta)}{p(\bbeta)}\,d\bbeta \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1568PICTUREFILESIZE 11348 $ Nv 8c?#"  R  d:is the Kullback-Leibler div. between posterior and prior. ;~ % # | A.8c?txp_fig#"  "8 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ D_{\mbox{\tiny Ber}}(E_{\mbox{\tiny emp}},E_{\mbox{\tiny gen}}) \leq \frac{1}{n}\left( D_{\mbox{\tiny KL}}(q||p) + \ln\frac{n+1}{\delta} \right) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2248PICTUREFILESIZE 11967H  0޽h ? ̙33___PPT10.+I ,D~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'%(+< 33# E3(      <| [ { k#Tightening the PAC-Bayesian Theorem$$ $    N | 8c?#"  8 Z0With our Laplacian prior and posterior, we have:1!l 4  # 4 ,$D 0   N | 8c?#"   4  8and show that @ 1   1    N| 8c?#"   kADue to the convexity of the KLD, it is easy to (numerically) findB>   # | A8c?txp_fig#"  ~H vn SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\boldeta}{\mbox{\boldmath$\eta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ D_{\mbox{\tiny min}} = \min_{\boldeta} D_{\mbox{\tiny KL}}(q||p)\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1066PICTUREFILESIZE4836(   # | A!8c?txp_fig#"   s 1 `X SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\boldeta}{\mbox{\boldmath$\eta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ D_{\mbox{\tiny min}} \leq \lambda\|\widehat{\bbeta}\|_1\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH706PICTUREFILESIZE3045   # | A$8c?txp_fig#"   SOURCEt\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ D_{\mbox{\tiny KL}}(q||p) = \sum_{i} \left[\ln \left( \frac{\eta_i}{e\, \lambda}\right) + \frac{\lambda}{\eta_i}\; e^{-\eta_i |\widehat{\beta_i}|} + \lambda |\widehat{\beta_i}|\right]. \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2428PICTUREFILESIZE 14012l D \  ! D \ ,$D 0   B A%8c?#"  D \    # | A8c?txp_fig#"  9o 6. SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \ln \left( \frac{\eta}{e\, \lambda}\right) + \frac{1}{\eta}\; e^{-\eta} + 1 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1066PICTUREFILESIZE5020#l   p "  p,$D 0   N<| 8c?#"     ISince we can choose freely:    # | A 8c?txp_fig#"   >  SOURCEd\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\boldeta}{\mbox{\boldmath$\eta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \boldeta\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE525   # | A-8c?txp_fig#"  = p& SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ D_{\mbox{\tiny Ber}}(E_{\mbox{\tiny emp}},E_{\mbox{\tiny gen}}) \leq \frac{1}{n}\left( \lambda\|\widehat{\bbeta}\|_1 + \ln\frac{n+1}{\delta} \right) = \mbox{bound}(\widehat\bbeta,{\cal D},\delta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH3018PICTUREFILESIZE 16112H   0޽h ? ̙33___PPT10.Нy+k?W D' = @B Du' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*# %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*! %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*" %(+^ eS]S4R(    <U| [* { fUsing the PAC-Bayesian Theorem    N[| 8c?#"  ` b8Set a prior parameter and choose a confidence level 9bl <0 b 1<0 b,$D 0   Na| 8c?#"  Bb @Find such that 6   # | A'8c?txp_fig#"  <0 Ynf SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\boldeta}{\mbox{\boldmath$\eta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ D_{\mbox{\tiny KL}}(q||p)\leq \lambda \|\widehat{\bbeta}\|_1 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH946PICTUREFILESIZE4887   # | A 8c?txp_fig#"  RI SOURCEd\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\boldeta}{\mbox{\boldmath$\eta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \boldeta\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE525ol ` 0`,$D 0  Nt| 8c?#"   [1Using this prior and , find a point estimate 2n  # | A8c?txp_fig#"  F& SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\cal D} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH104PICTUREFILESIZE668  # | A8c?txp_fig#"  `8 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\bbeta}. \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH114PICTUREFILESIZE985b * # | A&8c?txp_fig#"  B$ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \delta. \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE578b + # | A8c?txp_fig#"  J$ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \lambda \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE523>l    4  ,$D 0H ! # | A,8c?txp_fig#"  tH  x SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ D_{\mbox{\tiny Ber}}(E_{\mbox{\tiny emp}},E_{\mbox{\tiny gen}}) \leq \mbox{bound}(\widehat\bbeta,{\cal D},\delta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1696PICTUREFILESIZE8779 " Nt| 8c?#"     \2From these, we know that with probability at least3t - # | A8c?txp_fig#"    * SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ (1-\delta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH346PICTUREFILESIZE1340l   3 ,$D 0  NĮ| 8c?#"  p  nDand evaluate the corresponding by Monte Carlo. E . # | A*8c?txp_fig#"   |H  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \pagestyle{empty} \begin{document} \[ E_{\mbox{\tiny emp}}(q) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH366PICTUREFILESIZE2019r@  2F  Nl| 8c?#"   With this , define the  posterior as above, B  # | A 8c?txp_fig#"  /P SOURCEd\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\boldeta}{\mbox{\boldmath$\eta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \boldeta\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE525 / # | A 8c?txp_fig#"   N>6 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\boldeta}{\mbox{\boldmath$\eta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ q(\bbeta|\widehat{\bbeta},\boldeta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH476PICTUREFILESIZE3037H  0޽h ? ̙33J B ___PPT10" .4+D ' = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(+t XnPn "@m(    N| 8c?#"    HTo obtain an explicit bound on  <| [* { fUsing the PAC-Bayesian Theorem     # | A(8c?txp_fig#"   ; > SOURCE"\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ E_{\mbox{\tiny gen}} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH206PICTUREFILESIZE1030l _. 0  8. _0 ,$D 0 ( N| 8c?#"  . N J can easily be found numerically.!\ 1 # | A98c?txp_fig#"  m> , SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ b \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH54PICTUREFILESIZE475\ 2 # | A98c?txp_fig#"  _t SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ b \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH54PICTUREFILESIZE475 3 NT| 8c?#"    Fis always non-trivial, i.e.,n 5 # | A)8c?txp_fig#"   0 w$ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ b\leq 1. \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH306PICTUREFILESIZE1117Ol &&  @&& ,$D 0t && 6&&,$D 0H  # | A,8c?txp_fig#"  &&jx SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ D_{\mbox{\tiny Ber}}(E_{\mbox{\tiny emp}},E_{\mbox{\tiny gen}}) \leq \mbox{bound}(\widehat\bbeta,{\cal D},\delta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1696PICTUREFILESIZE8779r  # | A68c?txp_fig#"   * SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \Rightarrow \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH124PICTUREFILESIZE466 - # | A78c?txp_fig#"  V SOURCE:\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ E_{\mbox{\tiny gen}} \in [a,\, b] \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH596PICTUREFILESIZE2364 t e5   9e5  ,$D 0r % # | A68c?txp_fig#"  5 D * SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \Rightarrow \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH124PICTUREFILESIZE466 0 # | A:8c?txp_fig#"  e  J SOURCE.\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ E_{\mbox{\tiny gen}} \leq b \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH416PICTUREFILESIZE1885,t ?  ?? ,$D 0*t ?  7? ,$D 0  B A28c?#"    B  H D8c?". ` (` B B B D8c?"6X 6` B  B D8c?"h $p   # | A18c?txp_fig#"  D  > SOURCE"\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ E_{\mbox{\tiny emp}} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH226PICTUREFILESIZE1006  # | A+8c?txp_fig#"  ? n SOURCER\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \mbox{bound}(\widehat\bbeta,{\cal D},\delta) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH756PICTUREFILESIZE4173 ! # | A38c?txp_fig#"  [l SOURCEx\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ D_{\mbox{\tiny Ber}}(E_{\mbox{\tiny emp}},E_{\mbox{\tiny gen}}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH786PICTUREFILESIZE3757B " B D1?"Lf $f  $ # | A(8c?txp_fig#"   v > SOURCE"\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ E_{\mbox{\tiny gen}} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH206PICTUREFILESIZE1030Z . # | A88c?txp_fig#"   ~D  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ a\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH64PICTUREFILESIZE467\ / # | A98c?txp_fig#"   ee  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ b \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH54PICTUREFILESIZE475zB : < D8c?"k  zB = < D8c?"%k % zB > < D8c?"5k 5 H  0޽h ? ̙33___PPT10.0+]CD~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(+N   0  (    <Hg [* { fUsing the PAC-Bayesian Theorem    N`m 8c?#"  fl `Finally, notice that the PAC-Bayesian bound applies to the Gibbs classifier, but recall Lemma 2.a  Hp ?""p FLemma 2: for any  posterior , the generalization error of the BVC is less than twice that of the GC. &x mx   v A ?txp_fig" 80 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\boldeta}{\mbox{\boldmath$\eta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ q(\bbeta|\widehat{\bbeta},\boldeta)\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH476PICTUREFILESIZE3037  Nh 8c?#"  . ,$ 0 In practice, we have observed that the BVC usually generalizes as well as (often much better than) the GC. We believe the factor 2 can be reduced, but we have not yet been able to show it. H  0޽h ? ̙33___PPT10.7K+]%DO' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+  + @{(  r  < A#?"l  < [ { eExample of PAC-Bayesian Bound    Nč 8c?#"  *J H Mines dataset   # | A8c?txp_fig#"  Jv* SOURCEr\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\boldeta}{\mbox{\boldmath$\eta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ n\simeq 11000 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH526PICTUREFILESIZE1745   # | A/8c?txp_fig#"  Hi SOURCEf\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\boldeta}{\mbox{\boldmath$\eta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ d = 106 \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH406PICTUREFILESIZE1439   N 8c?#"  Q   n(Maybe tight enough to guide selection of)))b  # | A8c?txp_fig#"  <$ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \lambda \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH430(BOXHEIGHT301"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH74PICTUREFILESIZE523H  0޽h ? ̙33___PPT10i.df+D=' = @B +V P V(      <L [{,$ 0 _Conclusions for Part II  7   N$ 8c?#"  Y:,$ 0 q)PAC-Bayesian bound for sparse classifier.*-**A   NL 8c?#"  Lcl,$ 0 {3The bound (unlike VC bounds) is always non-trivial.4-44:   Nt 8c?#"  x<,$ 0 t,Tightness still requires large sample sizes.----M   N 8c?#"   ,$ 0 3Future goals: tightening the bounds, ..as always."&--44H   0޽h ? ̙3380___PPT10.v `D(     HHȉ ?" 4PART III Feature Selection in Unsupervised Learning554H  0޽h ? ̙33y___PPT10Y+D=' = @B + JBp (    <(ω  ?I 6 6FS is a widely studied problem in supervised learning.7 76  <xӉ  ?pI AFS is a not widely studied in unsupervised learning (clustering).B BAd  <׉  ?B,$ 0 vA good reason: in the absence of labels, how do you assess the usefulness of each feature?w wv  <t܉ kJ r*Feature Selection in Unsupervised Learning++ +    H ?" r ,$ 0 pApproach: how relevant is each feature (component of x) for the mixture nature of the data? :q  ,:ql ;   ; ,$D 0k  <  ?;,$ 0 YWe address this problem in the context of model-based clustering using finite mixtures: 8Z *Z    v A?txp_fig"   SOURCEl\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ p({\bf x})=\sum_{i=1}^k \alpha_i\, p({\bf x}|\btheta_i) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1066PICTUREFILESIZE7389H  0޽h ? ̙33___PPT10+ZD_D' = @B DF' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(+8+0+   +)  (  X  0 Ap ?Q   <D  ?pm [ Example:       <  ?@  6x2 is irrelevant for the mixture nature of this data.&7 56   <  ?  4x1 is relevant for the mixture nature of this data.&5 34 .  <  ?j J p ,$ 0 <Any PCA-type analysis of this data would not be useful here.= =<   <X k s+Example of Relevant and Irrelevant Features,, , H  0޽h ? ̙33___PPT10+T^DO' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+  + y (    B k y1Interplay Between Number of Clusters and Features22 2 ^  6 A~ ?   B  ?bv [ Example:     /  B$  ?"](,$ 0 %Using only x1, we find 2 components.&&  %   B\*  ? ,$ 0 XUsing x1 and x2, we find 7 components (needed to fit the non-Gaussian density of x2)JY GX   B1  ?lF \ Marginals     ~B  H DV?"e ~B   H DV?"8i H  0޽h ? ̙33s  @8(    <7  ?w]} RMost classical FS methods for supervised learning require combinatorial searches. S SR   <;  ?]87 6For d features, there are 2d possible feature subsets.&7 6   <A  ?],$ 0 Alternative: assign real valued feature weights and encourage sparseness (like seen above for supervised learning).  }    <,E k  gApproaches to Feature Selection      N\F 8c?#"   lU  y3[Law, Figueiredo and Jain, TPAMI, 2004 (to appear)]444H  0޽h ? ̙33___PPT10.]/ +T^DO' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+  +U OO9@3O(    <T [{ v.Maximum Likelihood Estimation and Missing Data// / nz _  8 _ ,$D 0  BY  ?_b Z Missing data     # | A?txp_fig" f SOURCEJ\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\cal Z} = \{{\bf z}_1,\dots,{\bf z}_n\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH866PICTUREFILESIZE2887  # | A?txp_fig"-td SOURCEH\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\bf z}_j = [z_j^{(1)},\dots,z_j^{(k)}] \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH936PICTUREFILESIZE4154,  # | A?txp_fig"X   jb SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{\bf x} \pagestyle{empty} \begin{document} \[ \Leftrightarrow \left\{\begin{array}{l}z_j^{(i)}=1\\ z_j^{(l)}=0 \Leftarrow l\neq i\end{array}\right. \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False*RESOLUTION600*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono0ORIGWIDTH109,8756PICTUREFILESIZE2878  Nv ?   U     B{  ?   7One-of-k encoding: component i &8 (7v # # | A?txp_fig"T  2 SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\bf x}_j \in \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH236PICTUREFILESIZE1125l [  9[ ,$D 00 /  v A?txp_fig"kk tl SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ L(\btheta,{\cal X},{\cal Z}) = \sum_{j=1}^n \sum_{i=1}^k z_j^{(i)}\log \alpha_i \, p({\bf x}_j|\btheta_i) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1968PICTUREFILESIZE 13662 0 <  ?[ i {  2 Complete log-likelihood  1 <  ?  v(...would be easy to maximize, if we had ) )(d 3  v A?txp_fig"j ( SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\cal Z} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH104PICTUREFILESIZE7338 Q 6Q   v A?txp_fig"M 2f SOURCEJ\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\cal X} = \{{\bf x}_1,\dots,{\bf x}_n\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH886PICTUREFILESIZE3069T    # *  H ?  U      <Э  ?   [ Training data     Bp  ?OA o lMaximum likelihood estimate of    v As?txp_fig" q$ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \btheta = \{\btheta_1,\dots,\btheta_k,\alpha_1,\dots,\alpha_{k-1}\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1526PICTUREFILESIZE4769 + 6  ?8 9 Swhere   ,  v A?txp_fig"! QPH SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ L(\btheta,{\cal X}) = \sum_{j=1}^n \log \sum_{i=1}^k \alpha_i \, p({\bf x}_j|\btheta_i) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1638PICTUREFILESIZE 11611 5  v A?txp_fig".#  SOURCEr\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\btheta} = \arg\max_{\btheta} L(\btheta,{\cal X}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1056PICTUREFILESIZE6094H  0޽h ? ̙33___PPT10./+ HD~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*9%(+p9 5z5, 5(    <Ս [{ v.EM Algorithm for Maximum Likelihood Estimation// /   <ڍ  ?[{ WE-step:     v A?txp_fig":   SOURCEl\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\cal W} = E[{\cal Z}|{\cal X},\widehat{\btheta}^{(t)}] \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH896PICTUREFILESIZE4827   v A?txp_fig"K  0n SOURCER\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\cal W} = \{ {\bf w}_1,\dots,{\bf w}_n \} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH966PICTUREFILESIZE3371   v A?txp_fig"+* @Vd SOURCEH\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\bf w}_j = [w_j^{(1)},\dots,w_j^{(k)}] \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1036PICTUREFILESIZE4616   6T  ?:{f  JCurrent estimate of the probability that was produced by component i"K IJd  # | A?txp_fig" am * SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\bf x}_j \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH114PICTUREFILESIZE738?l s = *o 9,$D 0  <L  ?  WM-step:  "  v A?txp_fig"s =4, SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \widehat{\btheta}^{(t+1)} = \arg\max_{\btheta} L(\btheta,{\cal X},{\cal W}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1436PICTUREFILESIZE8529  B0  ?O[I Xbecause   x  # | A?txp_fig"T< SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ z_j^{(i)}\in\{0,1\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH606PICTUREFILESIZE3009  H 8c?#" `GF + # | A8c?txp_fig" q  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ w_j^{(i)} = Pr(z_j^{(i)} = 1 |{\bf x}_j, \widehat{\btheta}^{(t)}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1336PICTUREFILESIZE7319xB , B D8c?"E}H  0޽h ? ̙33___PPT10n./7+%7DB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**%(+6 i)a)$((    <$' [ { dModel Selection for Mixtures    <+  ?!A 9Important issue: how to select k (number of components)? &: 9  <1  ?|h,$ 0 [Figueiredo & Jain, 2002] MML-based approach; roughly, an MDL/BIC with careful definition of the amount of data from which each parameter is estimated. l X  !X ,$D 0  <7  ? 8Resulting criterion leads to a simple modification of EM9 98   <46  ?E e  mOriginal M-step expression for       v A?txp_fig" X   SOURCEn\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \hat{\alpha}_i^{(t+1)} = \frac{1}{n}\sum_{j=1}^n w_j^{(i)}\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH936PICTUREFILESIZE6624   v A?txp_fig" ^ x D SOURCE(\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \hat{\alpha}_i^{(t+1)} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH286PICTUREFILESIZE1661l  f $f ,$D 0   <N  ? pj FThis update  kills weak components$ $#  B 8c?#" ` fl  b  " b ,$D 0  <DT  ?3 }S  ^New expression:     v A?txp_fig" b   SOURCEx\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \hat{\alpha}_i^{(t+1)} = \frac{\beta_i}{\sum_{l=1}^k \beta_l} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH846PICTUREFILESIZE5780  <`  ?   Swhere    v A?txp_fig"G t ,$ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \beta_i = \max\left\{\sum_{j=1}^n w_j^{(i)} -\frac{N_p}{2} ,0 \right\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1488PICTUREFILESIZE 10441l     #  ,$D 0  <l  ?    w'Number of parameters of each component.( ('    0e0e    BCDEFA 8c 8c8c     ?A)BCD|E||'$u @B@  c"$` q V H  0޽h ? ̙33  ___PPT10 ./l+OYD? ' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(+8+0+  +, ((_((    <z [ { fFeature Selection for Mixtures    <  ?SVY ISimplifying assumption: in each component, the features are independent. J JI   v A?txp_fig"8 2* SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ p({\bf x}|\btheta) = \sum_{i=1}^k \alpha_i\prod_{l=1}^d p(x_l|\theta_{il}) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1376PICTUREFILESIZE9787l  k% k %,$ 0-   <  ? k% =parameters of the density of the l-th feature in component i B> !=(   0e0e    BCDE(FA V 8c8c     ?A)BCD|E|| ~t:# @   c"$`  sl O O,$D 0   <  ? ,$ 0 Let some features have a common density in all components. These features are  irrelevant . ^ ^]t - D  - D ,$D 0   v A?txp_fig" Y L SOURCE0\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bphi}{\mbox{\boldmath$\phi$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bphi = \{\phi_1,...,\phi_d\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH806PICTUREFILESIZE3502  <  ? - M  yif feature l is relevant &    <  ?  7  {if feature l is irrelevant &     v A?txp_fig"L D  SOURCEf\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bphi}{\mbox{\boldmath$\phi$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \phi_l = \left\{\begin{array}{l}1,\\0,\end{array}\right. \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH486PICTUREFILESIZE2473u t Q O Q O,$D 0  <L  ? O 7Common (w.r.t. i) densities of the irrelevant features,8 (7B  B DV?" A P   v A?txp_fig"Q v  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ p({\bf x}|\btheta) = \sum_{i=1}^k \alpha_i\prod_{l=1}^d [p(x_l|\theta_{il})]^{\phi_l} \, [q(x_l|\lambda_{l})]^{1-\phi_l} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2168PICTUREFILESIZE 14250H  0޽h ? ̙33___PPT10n./2_+1DB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+7 1}1$ 1(    <×  ?Ss ]The likelihood  ;l B B,$D 0  <ȗ  ?Bb MTo apply EM, we see as missing data and define N NM   v A?txp_fig" [L SOURCE0\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bphi}{\mbox{\boldmath$\phi$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bphi = \{\phi_1,...,\phi_d\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH806PICTUREFILESIZE3502~    v A?txp_fig"[@ SOURCE$\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bphi}{\mbox{\boldmath$\phi$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \rho_l = Pr(\phi_l = 1) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH806PICTUREFILESIZE3162   <ۗ  ? :we call it  feature saliency    B  B B DV?"{{  < [{ XFeature Saliency     v A?txp_fig"'qr SOURCEV\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\bphi}{\mbox{\boldmath$\phi$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ p({\bf x}|\btheta,\bphi) = \sum_{i=1}^k \alpha_i\prod_{l=1}^d [p(x_l|\theta_{il})]^{\phi_l} \, [q(x_l|\lambda_{l})]^{1-\phi_l} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2298PICTUREFILESIZE 15233=l  ] $ ],$D 0   B  ?   CCan be shown that the resulting likelihood (marginal w.r.t. ) isD DCZ  # | A?txp_fig" k  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bphi}{\mbox{\boldmath$\phi$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bphi \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH94PICTUREFILESIZE694B  H DV?"f/ gk  # | A?txp_fig" nj ].& SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\brho}{\mbox{\boldmath$\rho$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \brho = \{\rho_1,...,\rho_l\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH756PICTUREFILESIZE2729 #  v A?txp_fig" G   SOURCEh\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\brho}{\mbox{\boldmath$\rho$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ p({\bf x}|\btheta,\brho) = \sum_{i=1}^k \alpha_i\prod_{l=1}^d \left[\rho_l\, p(x_l|\theta_{il}) + (1-\rho_l)\,q(x_l|\lambda_{l})\right] \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2608PICTUREFILESIZE 16062H  0޽h ? ̙33___PPT10./ +xPD~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(+4 00 !/(    < [{ S Applying EM      v A?txp_fig"~ SOURCEb\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\brho}{\mbox{\boldmath$\rho$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \left(\widehat{\btheta},\widehat{\brho}\right) = \arg\max_{\btheta,\brho}\left\{\sum_{j=1}^n\log p({\bf x}_j|\btheta,\brho) \right\} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1948PICTUREFILESIZE 14876   <  ?= X We address      <  ?  {-by EM, using and as missing data.. .-L    v A?txp_fig"# SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bphi}{\mbox{\boldmath$\phi$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \bphi \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH94PICTUREFILESIZE694T    v A?txp_fig">  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bphi}{\mbox{\boldmath$\phi$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ {\cal Z} \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH104PICTUREFILESIZE733rl W !W,$D 0F  B6  ?W]  lIn addition to the variables defined above, the E-step now also involves the following variables: &m :-ll  # | A?txp_fig"\( SOURCE \documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ w_j^{(i)}\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH186PICTUREFILESIZE1472  BC  ?  r$both easily computed in closed form.% %$  # | A48c?txp_fig#"  X  N SOURCE2\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\brho}{\mbox{\boldmath$\rho$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ v_{jl}^{(i)} = Pr\left( z_j^{(i)}=1,\phi_l = 0|{\bf x}_j,\widehat{\btheta}^{(t)},\widehat{\brho}^{(t)} \right) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1968PICTUREFILESIZE 11419  # | A08c?txp_fig#"  d { N SOURCE2\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\brho}{\mbox{\boldmath$\rho$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ u_{jl}^{(i)} = Pr\left( z_j^{(i)}=1,\phi_l = 1|{\bf x}_j,\widehat{\btheta}^{(t)},\widehat{\brho}^{(t)} \right) \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1968PICTUREFILESIZE 11219H  0޽h ? ̙33___PPT10n./3y+7CQDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(+UA g=_= , <(    <] [{ S Applying EM     <\b  ?  z"M-step: (ommitting the variances)#  "08    060  <Hh  ?  nAssuming that and are both univariate Gaussian with arbitrary mean and variance.o on    v A?txp_fig"    SOURCE|\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\brho}{\mbox{\boldmath$\rho$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ p(\cdot|\theta_{il})\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH336PICTUREFILESIZE2000    v A?txp_fig"  SOURCEx\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\brho}{\mbox{\boldmath$\rho$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ q(\cdot|\lambda_l)\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH316PICTUREFILESIZE1824   v A?txp_fig"J I  SOURCEn\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \hat{\alpha}_i^{(t+1)} = \frac{1}{n}\sum_{j=1}^n w_j^{(i)}\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH936PICTUREFILESIZE6624   v A?txp_fig"t A*" SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \hat{\rho}_l^{(t+1)} = \frac{1}{n}\sum_{j=1}^n \sum_{i=1}^k u_{jl}^{(i)}\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1096PICTUREFILESIZE7962l 7  ,7 ,$D 0  Bx  ?   Vmean in    ! # | A?txp_fig"  $ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\brho}{\mbox{\boldmath$\rho$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \hat{\theta}_{il}^{(t+1)}\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH266PICTUREFILESIZE1710 $ # | A?txp_fig"  SOURCE~\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\brho}{\mbox{\boldmath$\rho$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \hat{\lambda}_l^{(t+1)}\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH276PICTUREFILESIZE1600 ' Bx  ?  Vmean in    * # | A58c?txp_fig#"  7    SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ = \frac{\sum_{j=1}^n u_{jl}^{(i)}x_{jl}}{\sum_{j=1}^n u_{jl}^{(i)}}\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH746PICTUREFILESIZE7504$ + # | A;8c?txp_fig#"    \T SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ = \frac{\sum_{j=1}^n \sum_{i=1}^k v_{jl}^{(i)} x_{jl}}{ \sum_{j=1}^n \sum_{i=1}^k v_{jl}^{(i)}}\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1028PICTUREFILESIZE 10550H  0޽h ? ̙33___PPT10n./pyЄ+pDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(+b/ t+l+0*(    < [H{ WModel Selection    <  ?TZ dTo perform feature selection, we want to encourage some of the saliencies to become either 0 or 1. e ed  <Ģ  ?@ NThis can be achieved with the same MML-type criterion used above to select k &O KN'l e e,$D 0  <ʢ  ?e  gThe modified M-step is:      v A?txp_fig" RJ SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \hat{\rho}_l^{(t+1)} = \frac{\max\left\{\sum_{j}\sum_i u_{jl}^{(i)}-\frac{kR}{2},0\right\}}{ \max\left\{\sum_{j}\sum_i u_{jl}^{(i)}-\frac{kR}{2},0\right\}+ \max\left\{\sum_{j}\sum_i v_{jl}^{(i)}-\frac{S}{2},0\right\}}\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH3188PICTUREFILESIZE 24663   <,آ  ?   Vwhere:   R    v A?txp_fig" x  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ R \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH94PICTUREFILESIZE611R   v A?txp_fig"Xe SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ S \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH84PICTUREFILESIZE649B B B DV?"B B B DV?"= = @     Y   <H  ?   hnumber of parameters in     v A?txp_fig"   SOURCEh\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\brho}{\mbox{\boldmath$\rho$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \theta_{il}\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH114PICTUREFILESIZE831  B  ?  hnumber of parameters in     v A?txp_fig" 1U  SOURCEd\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\btheta}{\mbox{\boldmath$\theta$}} \newcommand{\brho}{\mbox{\boldmath$\rho$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \lambda_l\] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH104PICTUREFILESIZE675H  0޽h ? ̙33___PPT10n./p![+\ǂ8DB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+ *"@ (  Yl y y,$D 0f  6 A ?J o  B  ? - y3 H@8___PPT9  / samples, projected on the first two dimensions"  0/  @`D8 Z Z  <  ? >Mixture of 4 Gaussians (with identity covariance) with d = 10,? ?>   c $A ??En4x dAert  c $A ??ErHx dAert  c $A ??Eh Zx dAert  c $A ?? E Mx dAert   B  ?2. e2 relevant features     BD  ?K3 g8 irrelevant features  T   v A?txp_fig" w# SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \{ \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH54PICTUREFILESIZE389T   v A?txp_fig" I' SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \{ \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH54PICTUREFILESIZE389  <0 [4{ YSynthetic Example  H  0޽h ? ̙33___PPT10n./`Rx+ؠDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+ P (  r  6 ?#" `!VX  0 A ? q   0 A ?',$D 0  0 A ?r,$D 0  0 A ?e ,$D 0l & &,$D 0`  0 A ?&   <;  ? = ]  [final     <@  ?kM ]initial  x2   B  jJ?"`4    <D  ?N  aCommon density  H  0޽h ? ̙33J B ___PPT10" ./U%+>D ' = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+    `   (  f  0 A ?"`J  <O  ?Dfqd 5Feature saliency values (mean 1 s.d.) over 10 runs6 65 x2  B  V?"`mt   BpT  ?=E] grelevant features  x2   B  V?"` Y    B Y  ?    iirrelevant features     <t] [4{ YSynthetic Example  N   v A?txp_fig" f  SOURCE\documentclass[12pt]{article} \usepackage{amsmath,amssymb} \newcommand{\Bmat}{{\bf B}} \newcommand{\Hmat}{\bf H} \newcommand{\bbeta}{\mbox{\boldmath$\beta$}} \newcommand{\xv}{{\bf x}} \pagestyle{empty} \begin{document} \[ \pm \] \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False.TRANSPARENTTrue,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH451(BOXHEIGHT308"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono$ORIGWIDTH94PICTUREFILESIZE201H  0޽h ? ̙33___PPT10i./q+D=' = @B +! +#p')(    <i  ?h  y'- Several standard benchmark data sets:( (' Jz C # #"&##$$#C  6Dv  ?) k4  @`  6  ?) n19  @`  6  ?}  p4000  @`   6  ?}  Texture classification  @`   6  ?) k7  @`   6  ?) n18  @`   6H  ?}  p2320  @`   6  ?}  |Image segmentation  @`  6$S  ?) k2  @`  6X  ?) n30  @`  6Ѭ  ?}  o569  @`  6T۬  ?}  Wisconsin breast cancer  @`  6  ?)f k3  @`  6  ?f) n13  @`  6  ?} f o178  @`  6  ?f}  nWine  @`  6  ?)Cf kk  @`  6  ?C)f kd  @`  6,  ?} Cf kn  @`  60%  ?C} f pName  @`lB  < o ?CCfB  6 1 ?fffB  6 1 ?fB  6 1 ?fB  6 1 ?lB  < o ?lB   < o ?CfB ! 6 1 ?} C} fB " 6 1 ?CfB # 6 1 ?)C)lB $ < o ?Cl  d ) d,$D 0 % <*  ?   =- These are standard data-sets for supervised classification.> >=  & <,/  ?  '- We fit mixtures, ignoring the labels.( ''  ' <`4  ?D d 3- We classify the points and compare to the labels.4 43  ( <D9 [_{ Q Real Data   H  0޽h ? ̙33___PPT10n./0A+R$DB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(+ ".(    < [%{ ZReal Data: Results  =z u  # #"&NKKLKk   6  ?SS u   4.85 (0.98)$     @`  6lZ  ?6 S Su   4.04 (0.76)$     @`   68c  ?S 6 u  Texture classification  @`   64m  ?S0S   32.84 (5.10)$    @`   6g  ?6 0SS   20.19 (1.54)$    @`   6  ?06 S  |Image segmentation  @`   6p  ?S0  10.09 (2.70)$    @`  6\  ?6 S0  9.55 (1.99)$     @`  6  ?6 0 Wisconsin breast cancer  @`  6  ?S  8.06 (3.73)$     @`  6Գ  ?6 S  6.61 (3.91)$    @`  6̼  ?6  nWine  @`  6ح  ?S x % error (sd)     @`  6dϽ  ?6 S x % error (sd)     @`  6ؽ  ?6  pName  @`lB  < o ?fB  6 1 ?fB  6 1 ?fB  6 1 ?00fB  6 1 ?S S lB  < o ?u u lB  < o ?u fB  6 1 ?6 6 u fB  6 1 ?SSu lB  < o ?u    <ܽ  ?I vi Ywith FS   ! <  ?BDb \ without FS    F " <  ? n  v For these data-sets, our approach is able to improve the performance of mixture-based unsupervised classification.&; -< -wv H  0޽h ? ̙33___PPT10i./`$+D=' = @B +0 < 4   (    <誘[{ [Research Directions    HL?"ba "H  H?"`,$ 0 F More efficient algorithms for logistic regression with LASSO prior.G-GGs  HT?"CE,$ 0 q Investigating the performance of generalized Gaussian priors with exponents other than 1 (LASSO) or 2 (Ridge)r-rr6   H?" /- ,$ 0 |4 Deriving performce bounds for this type of approach5-55^l  7   7 ,$D 0  H?"& ^In supervised learning:    H?" 7  `In unsupervised learning:    H?" & ,$ 0 a More efficien algorithms-3   Hx?" /,$ 0 y1 Removing the conditional independence assumption2-22H   H?"N8]n,$ 0 F Extension to other mixtures (e.g., multinomial for categorical data).G-GGH  0޽h ? ̙33___PPT10t.N+AD' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(+P+0+n ++0+n ++0+ n ++0+ n ++0+ n ++0+ n +0Vx}`y&wN̵vq68U)U8$!)Ɍҝ^^鹖_ιfr$|yﵙ @D72o~yo/YʥG^|ZK W.-=,_|  y@<> yJ!w yd(W@V laKk!97@ys!|@ q o|!䭐/<bȗ@AC dB< yHR eHre/|SCNC@BAC.@< yZ}Zw!d r4ݐg B*sB|5}*&Y &d R܂܆4 ې&فiC:.d y?d5|y[8tH_zis^8/W}jnoݧOC_aS4xGãK?^F[^A?' 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Figueiredo%      .-@Times New Roman-. =2 r$Institute of Telecommunications, and         .-@Times New Roman-. .2 0Instituto Superior Tcnico      .-@Times New Roman-. 42 Technical University of Lisbon      .-@Times New Roman-. 2 PORTUGAL.-@Times New Roman-. U2 84Work herein reported was done in collaboration with:          .-@Times New Roman-. 02 kL. Carin, B. Krishnapuram, b    .-@Times New Roman-.  2 and .-@Times New Roman-. 2  A. Hartemink .-@Times New Roman-. "2 , Duke University;   .-@Times New Roman-. 2 k A. K. Jain k.-@Times New Roman-.  2 and .-@Times New Roman-. 2 M. 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