diff --git a/README.md b/README.md index 5f36fa7..fe404f9 100644 --- a/README.md +++ b/README.md @@ -28,9 +28,10 @@ If you find any issue, please let me know via [this](https://github.com/tiepvups + ## General Optimization problem - + where: @@ -40,8 +41,10 @@ where: ***Note***: this implementation also work on nonnegativity constrained problems. + ## Algorithms + ### If `L(f)` is easy to calculate, We use the following algorithm: ![FISTA with constant step](https://raw.githubusercontent.com/tiepvupsu/FISTA/master/figs/FISTA_L.png) @@ -50,6 +53,7 @@ where `pL(y)` is a proximal function defined as: For a new problem, our job is to implement two functions: `grad_f(x)` and `pL(y)` which are often simpler than the original optimization stated in (1). + ### In case `L(f)` is hard to find, We can alternatively use the following algorithm: (in this version, I haven't implemented this): @@ -57,6 +61,7 @@ We can alternatively use the following algorithm: (in this version, I haven't im where `QL(x, y)` is defined as: ![FISTA with backtracking](https://raw.githubusercontent.com/tiepvupsu/FISTA/master/figs/qlxy.png) + ## Usage `[X, iter, min_cost] = fista_general(grad, proj, Xinit, L, opts, calc_F) ` @@ -87,18 +92,20 @@ where: min_cost : the achieved cost ``` + ## Examples + ### Lasso (and weighted) problems ***Optimization problem:*** This function solves the l1 Lasso problem: - + if `lambda` is a scalar, or : - + if `lambda` is a matrix. In case `lambda` is a vector, it will be converted to a matrix with same columns and its # of columns = # of columns of `X`. @@ -275,17 +282,18 @@ FISTA provides a better cost. ``` + ### Elastic net problems ***Optimization problem:*** This function solves the [Elastic Net](https://web.stanford.edu/~hastie/Papers/B67.2%20(2005)%20301-320%20Zou%20&%20Hastie.pdf) problem: - + if `lambda` is a scalar, or : - + if `lambda` is a matrix. In case `lambda` is a vector, it will be convert to a matrix with same columns and its # of columns = # of columns of `X`. @@ -380,13 +388,14 @@ cost_fista = 6.10309e+00 cost_spams = 6.10309e+00 ``` + ### Row sparsity problems ***Optimization problem:*** - + -where and is the i-th row of . +where 'm' is number of rows and is the i-th row of . ***Matlab function:*** @@ -451,4 +460,5 @@ function test_row_sparsity() end ``` + ### Group sparsity problems (implement later) diff --git a/latex/definitions.tex b/latex/definitions.tex new file mode 100755 index 0000000..4128a3a --- /dev/null +++ b/latex/definitions.tex @@ -0,0 +1,240 @@ +\definecolor{colorsrc}{rgb}{0.36, 0.54, 0.66} + +% \definecolor{colornan}{rgb}{0.5, 0.5, 0.5} +% \definecolor{colornan}{rgb}{0.43, 0.21, 0.1} +% \definecolor{auburn}{rgb}{0.43, 0.21, 0.1} +% \definecolor{colorwnd2}{rgb}{1, .44, .37} +\definecolor{colorlcksvd}{rgb}{0.91, 0.84, 0.42} +\definecolor{colorlcksvdd}{rgb}{0.8, 0.0, 0.1} +\definecolor{colorlcksvd}{rgb}{1, 0.56, 0.0} +% \definecolor{colornan}{rgb}{0, 0.8, 0.0} +% \definecolor{colorsrc}{rgb}{0.5, 1, 0} +% \definecolor{colorfdd}{rgb}{0.6, 0.4, 0.8} +% \definecolor{colorfdd}{rgb}{0.93, 0.53, 0.18} +\definecolor{colorfddl}{rgb}{0.44, 0.16, 0.39} +\definecolor{colordlsi}{rgb}{0.55, 0.71, 0.0} +% \definecolor{colorlck}{rgb}{0.43, 0.21, 0.1} +% \definecolor{colorlck}{rgb}{0.89, 0.82, 0.04} +% \definecolor{colorlck}{rgb}{0.03, 0.27, 0.49} +\definecolor{colorcopar}{rgb}{0.9, .0, 0} +% \definecolor{colorlrsd}{rgb}{0.72, 0.53, 0.04} +\definecolor{darkcyan}{rgb}{0.0, 0.55, 0.55} +\definecolor{colordlr}{rgb}{0.0, 0.55, 0.55} +\definecolor{colorlrsdl}{rgb}{0.0, 0.2, 1.0} % blue +% \definecolor{colorlck}{rgb}{0.5, 0.5, 0.0} +% \definecolor{colorlck}{rgb}{0.0, 0.42, 0.24} +\definecolor{colorlck}{rgb}{0.0, 0.9, 0.9} +\definecolor{pinegreen}{rgb}{0.0, 0.47, 0.44} + +\def\myaddplotlrsdl{\addplot+[thick, colorlrsdl, solid, mark = square*, mark size=1.4, mark options={colorlrsdl}]} +\def\myaddplotfddl{\addplot+[thick, colorfddl, mark = diamond*, mark size=1.4, mark options={colorfddl}]} +\def\myaddplotlcksvd{\addplot+[thick, colorlcksvd, mark = x, mark size=1.8, mark options={colorlcksvd}]} +\def\myaddplotlcksvdd{\addplot+[thick, green, mark = triangle, mark size=1.4, mark options={green}]} +\def\myaddplotdlsi{\addplot+[thick, colordlsi, mark = *, mark size=1.4, mark options={fill = white}]} +\def\myaddplotsrc{\addplot+[thick, colorsrc, mark = diamond, mark size=1.4, mark options={colorsrc}]} +\def\myaddplotcopar{\addplot+[thick, colorcopar, solid, mark = *, mark size=1.4, mark options={colorcopar}]} +\def\myaddplotdlr{\addplot+[thick, colordlr, solid, dashed, mark = triangle*, mark size=1.4, mark options={colordlr}]} + + +% \def\myaddplotcopar{\addplot+[colorcopar, mark = square*, mark options={colorcopar}]} +% \def\myaddplotdfd{\addplot+[colordfd, mark options={colordfd}]} +% \def\myaddplotfdd{\addplot+[colorfdd, mark options={colorfdd}]} +% \def\myaddplotlck{\addplot+[colorlck, mark options={colorlck}]} +% \def\myaddplotnan{\addplot+[colornan, mark options={colornan}]} +% \def\myaddplotwnd{\addplot+[colorwnd, mark options={colorwnd}]} +% \def\wnd{{black,fill=colorwnd}} + +\def\x{{\mathbf x}} +\def\L{{\cal L}} + +\newcommand{\vect}[1]{\mathbf{#1}} + +\newcommand{\mat}[1]{\mathbf{#1}} +\newcommand{\abs}[1]{\left|#1\right|} +\newcommand{\norm}[1]{\left\|#1\right\|} +% \newcommand{\R}{\mathbb{R}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\tb}{\textbf} + + +\def\bmt{\left[\begin{matrix}} +\def\dpcm{$\square$} +\def\emt{\end{matrix}\right]} +% \def\proof{\underline{Proof:}\\} +\def\dpcm{$\square$} +\def\half{\frac{1}{2}} +\def\imply{\Rightarrow} +\def\foralli{\forall i = 1, 2, \dots, n} +\def\im{\mathrm{im}} +\def\ker{\mathrm{ker}} +\def\eqv{\Leftrightarrow} +\def\tcg{\textcolor{newgreen}} +\def\mb{\mathbf} +\def\tb{\textbf} +\def\mb {\mathbf} +\def\mc {\mathcal} +\def\tcb{\textcolor{blue}} +\def\tcg{\textcolor{green}} +\def\tcr{\textcolor{red}} +\def\tcgr{\textcolor{gray}} +\def\bx{\mathbf{x}} +% \def\bW{\mathbf{W}} +\def\ba{\mathbf{a}} +\def\bb{\mathbf{b}} +\def\bc{\mathbf{c}} +\def\bd{\mathbf{d}} +\def\be{\mathbf{e}} +\def\fb{\mathbf{f}} +\def\bg{\mathbf{g}} +\def\bh{\mathbf{h}} +\def\bm{\mathbf{m}} +\def\M{\mathcal{M}} +\def\bp{\mathbf{p}} +\def\bq{\mathbf{q}} +\def\bs{\mathbf{x}} +\def\bu{\mathbf{u}} +\def\bv{\mathbf{v}} +\def\by{\mathbf{y}} +\def\bz{\mathbf{z}} +\def\and{\text{~and~}} +\def\barN{\bar{N}} +\def\barNi{\bar{N}_i} +\def\trace{\textrm{trace}} +\def\etal{\textit{et al.}} +\def\R{\mathbb{R}} + +\def\bzeros{\mathbf{0}} + +\def\bA{\mathbf{A}} +\def\bB{\mathbf{B}} +\def\bD{\mathbf{D}} +\def\bE{\mathbf{E}} +\def\Fb{\mathbf{F}} +\def\bG{\mathbf{G}} +\def\bL{\mathbf{L}} +\def\bH{\mathbf{H}} +\def\bI{\mathbf{I}} +\def\bJ{\mathbf{J}} +\def\bM{\mathbf{M}} +\def\bN{\mathbf{N}} +\def\bP{\mathbf{P}} +\def\bQ{\mathbf{Q}} +\def\bR{\mathbf{R}} +\def\bS{\mathbf{S}} +\def\bU{\mathbf{U}} +\def\bV{\mathbf{V}} +\def\bW{\mathbf{W}} +\def\bX{\mathbf{X}} + +\def\bY{\mathbf{Y}} +\def\bZ{\mathbf{Z}} +\def\rank{\text{rank}} +\def\bDi{\mathbf{D}_i} +% \def\bSi{\mathbf{X}_i} +\def\bXi{\mathbf{X}_i} +% \def\bSi{\mathbf{X}_i} +\def\barX{\bar{\mathbf{X}}} +\def\barD{\bar{\mathbf{D}}} +\def\barX{\bar{\mathbf{X}}} +\def\barXi{\bar{\mathbf{X}}_i} +\def\barDi{\bar{\mathbf{D}}_i} +\def\barXi{\bar{\mathbf{X}}_i} +\def\bW{\mathbf{W}} +\def\bw{\mathbf{w}} + +\def\la{\langle} +\def\ra{\rangle} + +\def\bDc{\bD_{0}} +\def\bXc{\bX^{0}} +\def\mM{\mathcal{M}} +\def\wt{\widetilde} + +\def\bbX{\lbar{\bX}} +\def\bbx{\lbar{\bx}} +\def\bbY{\lbar{\bY}} +\def\bbD{\lbar{\bD}} + +%% ================== block: Slide footnotes ========================== +\def\footnoteSRC{\setcounter{footnote}{3}\footnote[frame]{\tiny J. Wright et al., Robust face recognition via sparse representation, IEEE TPAMI, 2009}} +\def\footnoteLLC{\setcounter{footnote}{4}\footnote[frame]{\tiny H. Zhang et. al., Locality-constrained linear coding for image classification, CPVR 2010}} +\def\footnoteJSRC{\setcounter{footnote}{5}\footnote[frame]{\tiny Yen, Multi-View Automatic Target Recognition using Joint Sparse Representation, Aerospace and Electronic Sys. 2012}} +\def\footnoteJDSRC{\setcounter{footnote}{6}\footnote[frame]{\tiny J. Wang et. al., Joint dynamic sparse representation for multi-view face recognition, PR 2012}} +\def\footnoteSHIRC{\setcounter{footnote}{7}\footnote[frame]{\tiny U. Srinivas et. al., Simultaneous sparsity model for histopathological image representation and classification, TMI 2014}} +\def\footnoteKSVD{\setcounter{footnote}{8}\footnote[frame]{\tiny M. Elad et. al., K -SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation, TSP 2006 }} +\def\footnoteODL{\setcounter{footnote}{9}\footnote[frame]{\tiny J. Mairal et. al., Online learning for matrix factorization and sparse coding, JMLR 2010}} +\def\footnoteDKSVD{\setcounter{footnote}{10}\footnote[frame]{\tiny Q. Zhang, B. Li, Discriminative K-SVD for dictionary learning in face recognition, CVPR 2010 }} + +\def\footnoteLCKSVD{\setcounter{footnote}{11}\footnote[frame]{\tiny Z. Jiang et. al., Label consistent K-SVD: Learning a discriminative dictionary for recognition, TPAMI 2013}} +\def\footnoteFDDL{\setcounter{footnote}{12}\footnote[frame]{\tiny M. Yang et. al., Fisher discrimination dictionary learning for sparse representation, ICCV 2011, IJCV 2014 }} +\def\footnoteDLR{\setcounter{footnote}{20}\footnote[frame]{\tiny L. Li et. al., Learning low-rank and discriminative dictionary for image classification, Image and Vision Computing, 2014}} +\def\footnoteOMP{\setcounter{footnote}{13}\footnote[frame]{\tiny Tropp et. al., Signal recovery from random measurements via orthogonal matching pursuit, IEEE Transactions on Information Theory 2007}} +\def\footnoteNANDITA{\setcounter{footnote}{14}\footnote[frame]{\tiny N. Nayak et. al., Classification of tumor histopathology via sparse feature learning, ISBI 2013}} +\def\footnoteWNDCHARM{\setcounter{footnote}{15}\footnote[frame]{\tiny L. Shamir et. al., WNDCHARM--an open source utility for biological image analysis, Source Code Biol. Med., 2008}} + +\def\footnoteDFDLTMI{\setcounter{footnote}{16}\footnote[frame]{\tiny \tcr{T. Vu} et. al., Histopathological Image Classification using Discriminative Feature-oriented Dictionary Learning, TMI 2015}} +\def\footnoteADMM{\setcounter{footnote}{17}\footnote[frame]{\tiny S. Boyd et. al., Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers, Foundations and Trends in Machine Learning, 2011}} +\def\footnoteFISTA{\setcounter{footnote}{18}\footnote[frame]{\tiny A. Beck et. al., A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM journal on Imaging sciences, 2009}} +\def\footnoteHojjatJPI{\setcounter{footnote}{19}\footnote[frame]{\tiny H. Mousavi \etal, Automated discrimination of lower and higher grade gliomas based on histopathological image analysis, JPI, 2015}} +% \def\footnote +% \def\footnoteDLSI{\setcounter{footnote}{3}\footnote[frame]{\tiny I. Ramirez et. al., IEEE Computer Vision and Pattern Recognition (CVPR), 2010 }} + + + + + + +\def\footnotea{\setcounter{footnote}{3}\footnote[frame]{\tiny H. Chang et. al., IEEE Transactions on Medical Imaging (TMI), 2013}} +\def\footnoteb{\setcounter{footnote}{4}\footnote[frame]{\tiny E. Ozdemir et. al., IEEE Transactions on Medical Imaging (TMI), 2013}} +\def\footnotec{\setcounter{footnote}{5}\footnote[frame]{\tiny M. Murat Dundar et. al., IEEE Transactions on Biomedical Engineering (TBME), 2011}} + +\def\footnoteGDDL{\setcounter{footnote}{21}\footnote[frame]{\tiny Suo et. al., Structured dictionary learning for classification, submitted to TSP 2014}} +\def\footnoteDLCORPA{\setcounter{footnote}{22}\footnote[frame]{\tiny S. Kong et. al., ECCV 2012}} +\def\footnoteDDLPC{\setcounter{footnote}{23}\footnote[frame]{\tiny Guo et. al., ACCV 2012}} +\def\footnoteLPDDL{\setcounter{footnote}{24}\footnote[frame]{\tiny Haghiri et. al., ICIP 2014}} +\def\footnoteDLSI{\setcounter{footnote}{25}\footnote[frame]{\tiny Ramirez et. al., Classification and clustering via dictionary learning with structured incoherence and shared features, CVPR 2010}} +\def\footnoteDFDL{\setcounter{footnote}{26}\footnote[frame]{\tiny T. Vu et. al., ISBI 2015}} +\def\footnoteDLRDSR{\setcounter{footnote}{27}\footnote[frame]{\tiny L. Ma et. al., CVPR 2012}} +\def\footnoteTDDL{\setcounter{footnote}{28}\footnote[frame]{\tiny J. Mairal et. al., ``Task-driven dictionary learning'', TPAMI, 2012}} +\def\footnoteTDDLLP{\setcounter{footnote}{29}\footnote[frame]{\tiny X. Sun, N. Nasrabadi, T. Tran, ``Task-driven dictionary learning for hyperspectral image classification with structured sparsity constraints'', TGRS, 2015}} +\def\footnoteJohn{\setcounter{footnote}{30}\footnote[frame]{\tiny J. Mckay, V. Monga \etal, Pose corrected sparsity for robust SONAR ATR, IGRSS, 2016}} + + + +\def\diag{\text{diag}} + + +%% ------------------end of block: Slide footnotes ---------------------------- + + + +\newcommand{\myFormA}[1]{\bmt #1 & \bzeros & \dots & \bzeros \\ \bzeros & #1 & \dots & \bzeros \\ \dots & \dots & \dots & \dots \\ \bzeros & \bzeros & \dots & #1 \emt} +\newcommand{\myFormB}[1]{\bmt #1 & #1 & \dots & #1 \\ #1 & #1 & \dots & #1 \\ \dots & \dots & \dots & \dots \\ #1 & #1 & \dots & #1 \emt} + + +%% ========= long bar notation ============================== +\makeatletter +\newsavebox\myboxA +\newsavebox\myboxB +\newlength\mylenA +\newcommand*\lbar[2][.75]{% + \sbox{\myboxA}{$\m@th#2$}% + \setbox\myboxB\null% Phantom box + \ht\myboxB=\ht\myboxA% + \dp\myboxB=\dp\myboxA% + \wd\myboxB=#1\wd\myboxA% Scale phantom + \sbox\myboxB{$\m@th\overline{\copy\myboxB}$}% Overlined phantom + \setlength\mylenA{\the\wd\myboxA}% calc width diff + \addtolength\mylenA{-\the\wd\myboxB}% + \ifdim\wd\myboxB<\wd\myboxA% + \rlap{\hskip 0.5\mylenA\usebox\myboxB}{\usebox\myboxA}% + \else + \hskip -0.3\mylenA\rlap{\usebox\myboxA}{\hskip 0.3\mylenA\usebox\myboxB}% + \fi} +\makeatother + +\def\lbD{\lbar{\bD}} +\def\lbY{\lbar{\bY}} +\def\lbX{\lbar{\bX}} +% \def\lbX{\overline{\mathbf{X}}} +% \def\lbY{\overline{\mathbf{Y}}} \ No newline at end of file diff --git a/latex/done b/latex/done new file mode 100644 index 0000000..82a0238 Binary files /dev/null and b/latex/done differ diff --git a/latex/fista1.aux b/latex/fista1.aux new file mode 100644 index 0000000..f23e546 --- /dev/null +++ b/latex/fista1.aux @@ -0,0 +1 @@ +\relax diff --git a/latex/fista1.fdb_latexmk b/latex/fista1.fdb_latexmk new file mode 100644 index 0000000..50b87ca --- /dev/null +++ b/latex/fista1.fdb_latexmk @@ -0,0 +1,206 @@ +# Fdb version 3 +["pdflatex"] 1485313677 "fista1.tex" "/home/tiepvu/myProjects/FISTA/latex/fista1.pdf" "fista1" 1485313677 + "/dev/null" 1485274908 0 d41d8cd98f00b204e9800998ecf8427e "" + 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