Matrix Factorization, Algorithms, Applications, and Avaliable packages

灰太狼 2022-09-20 15:29 166阅读 0赞

来源：[http://www.cvchina.info/2011/09/05/matrix-factorization-jungle/][http_www.cvchina.info_2011_09_05_matrix-factorization-jungle]

http://blog.csdn.net/abcjennifer/article/details/30371615

[Matrix Decompositions][] has a long history and generally centers around a set of known factorizations such as LU, QR, SVD and eigendecompositions. More [recent factorizations have seen the light of the day with work started with the advent of NMF, k-means and related algorithm][recent factorizations have seen the light of the day with work started with the advent of NMF_ k-means and related algorithm] \[[1][recent factorizations have seen the light of the day with work started with the advent of NMF_ k-means and related algorithm]\]. However, with the advent of new methods based on random projections and convex optimization that started in part in the [compressive sensing literature][], we are seeing another surge of very diverse algorithms dedicated to many different kinds of matrix factorizations with new constraints based on rank and/or positivity and/or sparsity,… As a result of this large increase in interest, I have decided to keep a list of them here following the success of the [big picture in compressive sensing][compressive sensing literature].

The sources for this list include the following most excellent sites: [Stephen Becker’s page][Stephen Becker_s page], [Raghunandan H. Keshavan][]‘ s [page][], [Nuclear Norm and Matrix Recovery][] through SDP by [Christoph Helmberg][], [Arvind Ganesh][]’s [Low-Rank Matrix Recovery and Completion via Convex Optimization][] who provide more in-depth additional information. Additional codes were featured also on [Nuit Blanche][]. The following people provided additional inputs: [Olivier Grisel][], [Matthieu Puigt][].

Most of the algorithms listed below generally rely on using the nuclear norm as a proxy to the rank functional. [It may not be optimal][]. Currently, [CVX][] ( [Michael Grant][] and [Stephen Boyd][]) consistently allows one to explore other proxies for the rank functional such as the[ log-det][log-det] as found by [Maryam Fazell][], [Haitham Hindi][], [Stephen Boyd][]. \*\* is used to show that the algorithm uses another heuristic than the nuclear norm.

In terms of notations, A refers to a matrix, L refers to a low rank matrix, S a sparse one and N to a noisy one. This page lists the different codes that implement the following matrix factorizations: Matrix Completion, Robust PCA , Noisy Robust PCA, Sparse PCA, NMF, Dictionary Learning, MMV, Randomized Algorithms and other factorizations. Some of these toolboxes can sometimes implement several of these decompositions and are listed accordingly. Before I list algorithm here, I generally feature them on Nuit Blanche under the MF tag: [http://nuit-blanche.blogspot.com/search/label/MF][http_nuit-blanche.blogspot.com_search_label_MF] or. [you can also subscribe to the Nuit Blanche feed,][you can also subscribe to the Nuit Blanche feed]

Matrix Completion, A = H.\*L with H a known mask, L unknown solve for L lowest rank possible

The idea of this approach is to complete the unknown coefficients of a matrix based on the fact that the matrix is low rank:

*  [OptSpace][]: [Matrix Completion from a Few Entries][] by [Raghunandan H. Keshavan][], [Andrea Montanari][], and [Sewoong Oh][]
 *  [LMaFit: Low-Rank Matrix Fitting][LMaFit_ Low-Rank Matrix Fitting]
 *  \*\* [Penalty Decomposition Methods for Rank Minimization][] by [Zhaosong Lu][] and [Yong Zhang][].The attendant [MATLAB code is here.][]
 *  [Jellyfish][]: [Parallel Stochastic Gradient Algorithms for Large-Scale Matrix Completion][], B. Recht, C. Re, Apr 2011
 *  [GROUSE][]: Online Identification and Tracking of Subspaces from Highly Incomplete Information, L. Balzano, R. Nowak, B. Recht, 2010
 *  [SVP][]: [Guaranteed Rank Minimization via Singular Value Projection][], R. Meka, P. Jain, I.S.Dhillon, 2009
 *  [SET][]: SET: an algorithm for consistent matrix completion, W. Dai, O. Milenkovic, 2009
 *  [NNLS][]: An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems, K. Toh, S. Yun, 2009
 *  [FPCA][]: Fixed point and Bregman iterative methods for matrix rank minimization, S. Ma, D. Goldfard, L. Chen, 2009
 *  [SVT][]: A singular value thresholding algorithm for matrix completion, J-F Cai, E.J. Candes, Z. Shen, 2008

Noisy Robust PCA, A = L + S + N with L, S, N unknown, solve for L low rank, S sparse, N noise

*  [GoDec][] : Randomized Low-rank and Sparse Matrix Decomposition in Noisy Case
 *  ReProCS: The [Recursive Projected Compressive Sensing code][] ([example][])

Robust PCA : A = L + S with L, S, N unknown, solve for L low rank, S sparse

*  [Robust PCA][] : Two Codes that go with the paper [“Two Proposals for Robust PCA Using Semidefinite Programming.”][Two Proposals for Robust PCA Using Semidefinite Programming.] by [MichaleI Mccoy][] and[ Joel Tropp][Joel Tropp]
 *  [SPAMS (SPArse Modeling Software)][SPAMS _SPArse Modeling Software]
 *  ADMM: [Alternating Direction Method of Multipliers][] [‘‘Fast Automatic Background Extraction via Robust PCA’][Fast Automatic Background Extraction via Robust PCA] by [Ivan Papusha][]. The [poster is here][]. The [matlab implementation is here][].
 *  PCP: [Generalized Principal Component Pursuit][]
 *  Augmented Lagrange Multiplier (ALM) Method \[exact ALM - MATLAB [zip][]\] \[inexact ALM - MATLAB[zip][zip 1]\], Reference - [The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices][], Z. Lin, M. Chen, L. Wu, and Y. Ma (UIUC Technical Report UILU-ENG-09-2215, November 2009)
 *  Accelerated Proximal Gradient , Reference - [Fast Convex Optimization Algorithms for Exact Recovery of a Corrupted Low-Rank Matrix][], Z. Lin, A. Ganesh, J. Wright, L. Wu, M. Chen, and Y. Ma (UIUC Technical Report UILU-ENG-09-2214, August 2009)\[full SVD version - MATLAB [zip][zip 2]\] \[partial SVD version - MATLAB [zip][zip 3]\]
 *  Dual Method \[MATLAB [zip][zip 4]\], Reference - [Fast Convex Optimization Algorithms for Exact Recovery of a Corrupted Low-Rank Matrix][], Z. Lin, A. Ganesh, J. Wright, L. Wu, M. Chen, and Y. Ma (UIUC Technical Report UILU-ENG-09-2214, August 2009).
 *  Singular Value Thresholding \[MATLAB [zip][zip 5]\]. Reference - [A Singular Value Thresholding Algorithm for Matrix Completion][], J. -F. Cai, E. J. Candès, and Z. Shen (2008).
 *  Alternating Direction Method \[MATLAB [zip][zip 6]\] , Reference - [Sparse and Low-Rank Matrix Decomposition via Alternating Direction Methods][], X. Yuan, and J. Yang (2009).
 *  [LMaFit: Low-Rank Matrix Fitting][LMaFit_ Low-Rank Matrix Fitting]
 *  [Bayesian robust PCA][]
 *  [Compressive-Projection PCA][] (CPPCA)

Sparse PCA: A = DX with unknown D and X, solve for sparse D

Sparse PCA [on wikipedia][]

*  R. Jenatton, G. Obozinski, F. Bach. Structured Sparse Principal Component Analysis. International Conference on Artificial Intelligence and Statistics (AISTATS). \[[pdf][]\] \[[code][]\]
 *  [SPAMs][SPAMS _SPArse Modeling Software]
 *  DSPCA: [Sparse PCA using SDP][] . Code is[ here][here].
 *  PathPCA: A fast greedy algorithm for Sparse PCA. The code is [here][here 1].

Dictionary Learning: A = DX with unknown D and X, solve for sparse X

Some implementation of dictionary learning implement the NMF

*  [Online Learning for Matrix Factorization and Sparse Coding][] by [Julien Mairal][], [Francis Bach][], [Jean Ponce][],[Guillermo Sapiro][] \[The code is released as [SPArse Modeling Software][]or [SPAMS][SPArse Modeling Software]\]
 *  [Dictionary Learning Algorithms for Sparse Representation][] (Matlab implementation of [FOCUSS/FOCUSS-CNDL is here][FOCUSS_FOCUSS-CNDL is here])
 *  [Multiscale sparse image representation with learned dictionaries][] \[Matlab implementation of the [K-SVD algorithm is here][], a newer implementation by Ron Rubinstein is [here][here 2] \]
 *  [Efficient sparse coding algorithms][] \[ Matlab [code is here][] \]
 *  [Shift Invariant Sparse Coding of Image and Music Data][]. Matlab implemention is [here][here 3]
 *  [Shift-invariant dictionary learning for sparse representations: extending K-SVD][Shift-invariant dictionary learning for sparse representations_ extending K-SVD].
 *  [Thresholded Smoothed-L0 (SL0) Dictionary Learning for Sparse Representations][Thresholded Smoothed-L0 _SL0_ Dictionary Learning for Sparse Representations] by Hadi Zayyani, [Massoud Babaie-Zadeh][] and [Remi Gribonval][].
 *  [Non-negative Sparse Modeling of Textures (NMF)][Non-negative Sparse Modeling of Textures _NMF] \[Matlab implementation of [NMF (Non-negative Matrix Factorization) and NTF (Non-negative Tensor][NMF _Non-negative Matrix Factorization_ and NTF _Non-negative Tensor]), a faster implementation of NMF can be found [here][here 4], here is a more recent [Non-Negative Tensor Factorizations package][]\]

NMF: A = DX with unknown D and X, solve for elements of D,X > 0

[Non-negative Matrix Factorization (NMF) on wikipedia][Non-negative Matrix Factorization _NMF_ on wikipedia]

*  [HALS][]: [Accelerated Multiplicative Updates and Hierarchical ALS Algorithms for Nonnegative Matrix Factorization][] by [Nicolas Gillis][], [François Glineur][Fran_ois Glineur].
 *  [SPAMS (SPArse Modeling Software)][SPAMS _SPArse Modeling Software] by [Julien Mairal][], [Francis Bach][], [Jean Ponce][],[Guillermo Sapiro][]
 *  [NMF][]: C.-J. Lin. [Projected gradient methods for non-negative matrix factorization][]. [Neural Computation][], 19(2007), 2756-2779.
 *  [Non-Negative Matrix Factorization][]: [This page][Non-Negative Matrix Factorization] contains an optimized C implementation of the Non-Negative Matrix Factorization (NMF) algorithm, described in [\[Lee & Seung 2001\]][Lee _ Seung 2001]. We implement the update rules that minimize a weighted SSD error metric. A detailed description of weighted NMF can be found in[\[Peers et al. 2006\]][Peers et al. 2006].
 *  [NTFLAB for Signal Processing][NMF _Non-negative Matrix Factorization_ and NTF _Non-negative Tensor], Toolboxes for NMF (Non-negative Matrix Factorization) and NTF (Non-negative Tensor Factorization) for BSS (Blind Source Separation)
 *  [Non-negative Sparse Modeling of Textures (NMF)][Non-negative Sparse Modeling of Textures _NMF] \[Matlab implementation of [NMF (Non-negative Matrix Factorization) and NTF (Non-negative Tensor][NMF _Non-negative Matrix Factorization_ and NTF _Non-negative Tensor]), a faster implementation of NMF can be found [here][here 4], here is a more recent [Non-Negative Tensor Factorizations package][]\]

Multiple Measurement Vector (MMV) Y = A X with unknown X and rows of X are sparse.

*  [T-MSBL/T-SBL][T-MSBL_T-SBL] by [Zhilin Zhang][]
 *  [Compressive MUSIC with optimized partial support for joint sparse recovery][] by [Jong Min Kim][], [Ok Kyun Lee][Jong Min Kim], [Jong Chul Ye][] \[no code\]
 *  [The REMBO Algorithm Accelerated Recovery of Jointly Sparse Vectors ][The REMBO Algorithm Accelerated Recovery of Jointly Sparse Vectors]by Moshe Mishali and Yonina C. Eldar \[ no code\]

Blind Source Separation (BSS) Y = A X with unknown A and X and statistical independence between columns of X or subspaces of columns of X

Include Independent Component Analysis (ICA), Independent Subspace Analysis (ISA), and Sparse Component Analysis (SCA). There are many available codes for ICA and some for SCA. Here is a non-exhaustive list of some famous ones (which are not limited to linear instantaneous mixtures). TBC

ICA:

*  ICALab: [http://www.bsp.brain.riken.jp/ICALAB/][http_www.bsp.brain.riken.jp_ICALAB]
 *  BLISS softwares: [http://www.lis.inpg.fr/pages\_perso/bliss/deliverables/d20.html][http_www.lis.inpg.fr_pages_perso_bliss_deliverables_d20.html]
 *  MISEP: [http://www.lx.it.pt/~lbalmeida/ica/mitoolbox.html][http_www.lx.it.pt_lbalmeida_ica_mitoolbox.html]
 *  Parra and Spence’s frequency-domain convolutive ICA:[http://people.kyb.tuebingen.mpg.de/harmeling/code/convbss-0.1.tar][http_people.kyb.tuebingen.mpg.de_harmeling_code_convbss-0.1.tar]
 *  C-FICA: [http://www.ast.obs-mip.fr/c-fica][http_www.ast.obs-mip.fr_c-fica]

SCA:

*  DUET: [http://sparse.ucd.ie/publications/rickard07duet.pdf][http_sparse.ucd.ie_publications_rickard07duet.pdf] (the matlab code is given at the end of this pdf document)
 *  LI-TIFROM: [http://www.ast.obs-mip.fr/li-tifrom][http_www.ast.obs-mip.fr_li-tifrom]

Randomized Algorithms

These algorithms uses generally random projections to shrink very large problems into smaller ones that can be amenable to traditional matrix factorization methods.

Resource  
[Randomized algorithms for matrices and data][] by Michael W. Mahoney  
[Randomized Algorithms for Low-Rank Matrix Decomposition][]

*  [Randomized PCA][]
 *  Randomized Least Squares: [Blendenpik][]( [http://pdos.csail.mit.edu/~petar/papers/blendenpik-v1.pdf][http_pdos.csail.mit.edu_petar_papers_blendenpik-v1.pdf] )

Other factorization

D(T(.)) = L + E with unknown L, E and unknown transformation T and solve for transformation T, Low Rank L and Noise E

*  [RASL: Robust Batch Alignment of Images by Sparse and Low-Rank Decomposition][RASL_ Robust Batch Alignment of Images by Sparse and Low-Rank Decomposition]
 *  [TILT: Transform Invariant Low-rank Textures][TILT_ Transform Invariant Low-rank Textures]

Frameworks featuring advanced Matrix factorizations

For the time being, few have integrated the most recent factorizations.

*  [Scikit Learn][] (Python)
 *  [Matlab Toolbox for Dimensionality Reduction][] (Probabilistic PCA, Factor Analysis (FA)…)
 *  [Orange][] (Python)
 *  [pcaMethods][]—a [bioconductor][] package providing PCA methods for incomplete data. R Language

GraphLab / Hadoop

*  [Danny Bickson][] keeps a [blog on GraphLab][].

Books

*  [Matrix Factorizations on Amazon][].

Example of use

*  [CS: Low Rank Compressive Spectral Imaging and a multishot CASSI][CS_ Low Rank Compressive Spectral Imaging and a multishot CASSI]
 *  [CS: Heuristics for Rank Proxy and how it changes everything….][CS_ Heuristics for Rank Proxy and how it changes everything_.]
 *  [Tennis Players are Sparse !][Alternating Direction Method of Multipliers]

Sources

[Arvind Ganesh][]’s [Low-Rank Matrix Recovery and Completion via Convex Optimization][]

*  [Raghunandan H. Keshavan][]‘ s [list][page]
 *  [Stephen Becker’s list][Stephen Becker_s page]
 *  [Nuclear Norm and Matrix Recovery][] through SDP by [Christoph Helmberg][]
 *  [Nuit Blanche][]

Relevant links

*  [Welcome to the Matrix Factorization Jungle][]
 *  [Finding Structure With Randomness][]

Reference:

[A Uniﬁed View of Matrix Factorization Models by Ajit P. Singh and Geoffrey J. Gordon][recent factorizations have seen the light of the day with work started with the advent of NMF_ k-means and related algorithm]

本文引用地址：[http://blog.sciencenet.cn/blog-242887-483128.html ][http_blog.sciencenet.cn_blog-242887-483128.html]

[http_www.cvchina.info_2011_09_05_matrix-factorization-jungle]: http://www.cvchina.info/2011/09/05/matrix-factorization-jungle/
[Matrix Decompositions]: http://en.wikipedia.org/wiki/Matrix_decomposition
[recent factorizations have seen the light of the day with work started with the advent of NMF_ k-means and related algorithm]: http://www.cs.cmu.edu/~ggordon/singh-gordon-unified-factorization-ecml.pdf
[compressive sensing literature]: https://sites.google.com/site/igorcarron2/cs
[Stephen Becker_s page]: http://www.ugcs.caltech.edu/~srbecker/algorithms.shtml
[Raghunandan H. Keshavan]: http://www.stanford.edu/~raghuram
[page]: http://www.stanford.edu/~raghuram/optspace/papers.html
[Nuclear Norm and Matrix Recovery]: http://www-user.tu-chemnitz.de/~helmberg/sdp_nuc.html
[Christoph Helmberg]: http://www-user.tu-chemnitz.de/~helmberg/
[Arvind Ganesh]: https://netfiles.uiuc.edu/abalasu2/www/home.html
[Low-Rank Matrix Recovery and Completion via Convex Optimization]: http://perception.csl.uiuc.edu/matrix-rank/
[Nuit Blanche]: http://nuit-blanche.blogspot.com/
[Olivier Grisel]: http://oliviergrisel.name/
[Matthieu Puigt]: http://www.ics.forth.gr/~mpuigt/
[It may not be optimal]: http://nuit-blanche.blogspot.com/2011/08/all-is-not-well-with-nuclear-norm-world.html
[CVX]: http://cvxr.com/cvx/
[Michael Grant]: http://cvxr.com/
[Stephen Boyd]: http://www.stanford.edu/~boyd/
[log-det]: http://www.stanford.edu/~boyd/papers/pdf/rank_min_heur_hankel.pdf
[Maryam Fazell]: http://faculty.washington.edu/mfazel/
[Haitham Hindi]: https://sites.google.com/site/hhindi1969/
[http_nuit-blanche.blogspot.com_search_label_MF]: http://nuit-blanche.blogspot.com/search/label/MF
[you can also subscribe to the Nuit Blanche feed]: http://feeds.feedburner.com/blogspot/wCeDd
[OptSpace]: http://www.stanford.edu/~raghuram/optspace/index.html
[Matrix Completion from a Few Entries]: http://arxiv.org/pdf/0901.3150
[Andrea Montanari]: http://www.stanford.edu/~montanar
[Sewoong Oh]: http://www.stanford.edu/~swoh
[LMaFit_ Low-Rank Matrix Fitting]: http://lmafit.blogs.rice.edu/
[Penalty Decomposition Methods for Rank Minimization]: http://www.math.sfu.ca/~zhaosong/ResearchPapers/pd-rank.pdf
[Zhaosong Lu]: http://people.math.sfu.ca/~zhaosong/
[Yong Zhang]: http://www.sfu.ca/~yza30/
[MATLAB code is here.]: http://www.math.sfu.ca/~zhaosong/Codes/PD-rank
[Jellyfish]: http://pages.cs.wisc.edu/~chrisre/Jellyfish.tbz2
[Parallel Stochastic Gradient Algorithms for Large-Scale Matrix Completion]: http://www.optimization-online.org/DB_HTML/2011/04/3012.html
[GROUSE]: http://sunbeam.ece.wisc.edu/research_misc.html
[SVP]: http://userweb.cs.utexas.edu/~pjain/svp/
[Guaranteed Rank Minimization via Singular Value Projection]: http://arxiv.org/abs/0909.5457
[SET]: http://watt.csl.illinois.edu/~perceive/matrix-rank/sample_code.html#MC
[NNLS]: http://www.math.nus.edu.sg/~mattohkc/NNLS.html
[FPCA]: http://www.columbia.edu/~sm2756/FPCA.htm
[SVT]: http://svt.caltech.edu/
[GoDec]: https://sites.google.com/site/godecomposition/
[Recursive Projected Compressive Sensing code]: http://home.engineering.iastate.edu/~chenlu/ReProCS/ReProCS_main.htm
[example]: http://home.engineering.iastate.edu/~chenlu/ReProCS/ReProCS.htm
[Robust PCA]: http://users.cms.caltech.edu/~mccoy/code/
[Two Proposals for Robust PCA Using Semidefinite Programming.]: http://users.cms.caltech.edu/~mccoy/publications/robustPCA_final.pdf
[MichaleI Mccoy]: http://users.cms.caltech.edu/~mccoy/
[Joel Tropp]: http://users.cms.caltech.edu/~jtropp/
[SPAMS _SPArse Modeling Software]: http://www.di.ens.fr/willow/SPAMS/index.html
[Alternating Direction Method of Multipliers]: http://nuit-blanche.blogspot.com/2011/08/tennis-players-are-sparse.html
[Fast Automatic Background Extraction via Robust PCA]: http://www.stanford.edu/~ipapusha/papers/ee364b_paper.pdf
[Ivan Papusha]: http://www.stanford.edu/~ipapusha/
[poster is here]: http://www.stanford.edu/~ipapusha/papers/ee364b_poster.pdf
[matlab implementation is here]: http://www.stanford.edu/~ipapusha/papers/pcp_admm.m
[Generalized Principal Component Pursuit]: http://www.mathworks.com/matlabcentral/fileexchange/28677
[zip]: http://perception.csl.uiuc.edu/matrix-rank/Files/exact_alm_rpca.zip
[zip 1]: http://perception.csl.uiuc.edu/matrix-rank/Files/inexact_alm_rpca.zip
[The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices]: http://perception.csl.uiuc.edu/matrix-rank/Files/Lin09-MP.pdf
[Fast Convex Optimization Algorithms for Exact Recovery of a Corrupted Low-Rank Matrix]: http://perception.csl.uiuc.edu/matrix-rank/Files/rpca_algorithms.pdf
[zip 2]: http://perception.csl.uiuc.edu/matrix-rank/Files/apg.zip
[zip 3]: http://perception.csl.uiuc.edu/matrix-rank/Files/apg_partial.zip
[zip 4]: http://perception.csl.uiuc.edu/matrix-rank/Files/dual.zip
[zip 5]: http://perception.csl.uiuc.edu/matrix-rank/Files/svt.zip
[A Singular Value Thresholding Algorithm for Matrix Completion]: http://www.acm.caltech.edu/~emmanuel/papers/SVT.pdf
[zip 6]: http://perception.csl.uiuc.edu/matrix-rank/Files/LRSD-Code.zip
[Sparse and Low-Rank Matrix Decomposition via Alternating Direction Methods]: http://www.optimization-online.org/DB_HTML/2009/11/2447.html
[Bayesian robust PCA]: http://people.ee.duke.edu/~lcarin/BCS.html
[Compressive-Projection PCA]: http://www.ece.msstate.edu/~fowler/CPPCA/
[on wikipedia]: http://en.wikipedia.org/wiki/Sparse_PCA
[pdf]: http://www.di.ens.fr/~jenatton/paper/sspca_AISTATS2010.pdf
[code]: http://www.di.ens.fr/~jenatton/software/SparseStructuredPCA_MatlabToolbox_V1.0_rjenatton.tar
[Sparse PCA using SDP]: http://www.princeton.edu/~aspremon/NIPS04.pdf
[here]: http://www.cmap.polytechnique.fr/~aspremon/DSPCA.html
[here 1]: http://www.cmap.polytechnique.fr/~aspremon/PathSPCA.html
[Online Learning for Matrix Factorization and Sparse Coding]: https://docs.google.com/viewer?url=http%3A%2F%2Fhal.inria.fr%2Fdocs%2F00%2F40%2F87%2F16%2FPDF%2Fmain.pdf
[Julien Mairal]: http://www.di.ens.fr/~mairal/
[Francis Bach]: http://www.di.ens.fr/~fbach/
[Jean Ponce]: http://www.di.ens.fr/~ponce/
[Guillermo Sapiro]: http://www.ece.umn.edu/~guille/
[SPArse Modeling Software]: http://www.di.ens.fr/willow/SPAMS/
[Dictionary Learning Algorithms for Sparse Representation]: https://docs.google.com/viewer?url=http%3A%2F%2Fdsp.ucsd.edu%2F%7Ekreutz%2FKreutz2003.pdf
[FOCUSS_FOCUSS-CNDL is here]: http://hebb.mit.edu/people/jfmurray/software.htm
[Multiscale sparse image representation with learned dictionaries]: https://docs.google.com/viewer?url=http%3A%2F%2Fwww.ima.umn.edu%2Fpreprints%2Fjan2007%2F2152.pdf
[K-SVD algorithm is here]: http://www.cs.technion.ac.il/~elad/software/
[here 2]: http://www.cs.technion.ac.il/~ronrubin/software.html
[Efficient sparse coding algorithms]: https://docs.google.com/viewer?url=http%3A%2F%2Fwww.stanford.edu%2F%7Ehllee%2Fnips06-sparsecoding.pdf
[code is here]: http://www.stanford.edu/~hllee/softwares/nips06-sparsecoding.htm
[Shift Invariant Sparse Coding of Image and Music Data]: https://docs.google.com/viewer?url=http%3A%2F%2Fwww2.imm.dtu.dk%2Fpubdb%2Fviews%2Fedoc_download.php%2F4659%2Fpdf%2Fimm4659.pdf
[here 3]: http://mikkelschmidt.dk/index.php?id=23&tx_ttnews%5Btt_news%5D=30&tx_ttnews%5BbackPid%5D=22&cHash=8c899eb7f5
[Shift-invariant dictionary learning for sparse representations_ extending K-SVD]: https://docs.google.com/viewer?url=http%3A%2F%2Fwww.irisa.fr%2Fmetiss%2Fgribonval%2FConf%2F2008%2Feusipco_shiftinvariantksvd_final.pdf
[Thresholded Smoothed-L0 _SL0_ Dictionary Learning for Sparse Representations]: https://docs.google.com/viewer?url=http%3A%2F%2Fwww.irisa.fr%2Fmetiss%2Fgribonval%2FConf%2F2009%2F2009_ICASSP_ZayyaniEtAl_SL0DicoLearning-final.pdf
[Massoud Babaie-Zadeh]: http://sharif.ir/~mbzadeh/
[Remi Gribonval]: http://www.irisa.fr/metiss/gribonval/
[Non-negative Sparse Modeling of Textures _NMF]: https://docs.google.com/viewer?url=http%3A%2F%2Fwww.ceremade.dauphine.fr%2F%7Epeyre%2Fpublications%2F%2F07-SSVM-Peyre-SynthesisSparsity.pdf
[NMF _Non-negative Matrix Factorization_ and NTF _Non-negative Tensor]: http://www.bsp.brain.riken.jp/ICALAB/nmflab.html
[here 4]: http://www.csie.ntu.edu.tw/~cjlin/nmf/index.html
[Non-Negative Tensor Factorizations package]: http://www.cs.ubc.ca/~mpf/?n=Papers.Lsntf
[Non-negative Matrix Factorization _NMF_ on wikipedia]: http://en.wikipedia.org/wiki/Non-negative_matrix_factorization
[HALS]: http://www.core.ucl.ac.be/~ngillis/papers/Acc_MU_HALS_PG.zip
[Accelerated Multiplicative Updates and Hierarchical ALS Algorithms for Nonnegative Matrix Factorization]: http://arxiv.org/PS_cache/arxiv/pdf/1107/1107.5194v1.pdf
[Nicolas Gillis]: http://www.core.ucl.ac.be/~ngillis/
[Fran_ois Glineur]: http://www.core.ucl.ac.be/~glineur/
[NMF]: http://www.csie.ntu.edu.tw/~cjlin/nmf/
[Projected gradient methods for non-negative matrix factorization]: http://www.csie.ntu.edu.tw/~cjlin/papers/pgradnmf.pdf
[Neural Computation]: http://mitpress.mit.edu/journal-home.tcl?issn=08997667
[Non-Negative Matrix Factorization]: http://www.cs.virginia.edu/~jdl/nmf/
[Lee _ Seung 2001]: http://scholar.google.com/url?sa=U&q=http://hebb.mit.edu/people/seung/papers/nmfconverge.pdf
[Peers et al. 2006]: http://www.cs.kuleuven.ac.be/publicaties/rapporten/cw/CW440.pdf
[T-MSBL_T-SBL]: http://dsp.ucsd.edu/~zhilin/TMSBL.html
[Zhilin Zhang]: http://dsp.ucsd.edu/~zhilin/
[Compressive MUSIC with optimized partial support for joint sparse recovery]: http://arxiv.org/PS_cache/arxiv/pdf/1102/1102.3288v1.pdf
[Jong Min Kim]: http://bisp.kaist.ac.kr/members.html
[Jong Chul Ye]: http://bisp.kaist.ac.kr/professor.htm
[The REMBO Algorithm Accelerated Recovery of Jointly Sparse Vectors]: http://webee.technion.ac.il/Sites/People/YoninaEldar/Info/1569104696.pdf
[http_www.bsp.brain.riken.jp_ICALAB]: http://www.linkedin.com/redirect?url=http%3A%2F%2Fwww%2Ebsp%2Ebrain%2Eriken%2Ejp%2FICALAB%2F&urlhash=8N_s&_t=tracking_disc
[http_www.lis.inpg.fr_pages_perso_bliss_deliverables_d20.html]: http://www.linkedin.com/redirect?url=http%3A%2F%2Fwww%2Elis%2Einpg%2Efr%2Fpages_perso%2Fbliss%2Fdeliverables%2Fd20%2Ehtml&urlhash=5cMy&_t=tracking_disc
[http_www.lx.it.pt_lbalmeida_ica_mitoolbox.html]: http://www.linkedin.com/redirect?url=http%3A%2F%2Fwww%2Elx%2Eit%2Ept%2F%7Elbalmeida%2Fica%2Fmitoolbox%2Ehtml&urlhash=ck5c&_t=tracking_disc
[http_people.kyb.tuebingen.mpg.de_harmeling_code_convbss-0.1.tar]: http://www.linkedin.com/redirect?url=http%3A%2F%2Fpeople%2Ekyb%2Etuebingen%2Empg%2Ede%2Fharmeling%2Fcode%2Fconvbss-0%2E1%2Etar&urlhash=QG6U&_t=tracking_disc
[http_www.ast.obs-mip.fr_c-fica]: http://www.linkedin.com/redirect?url=http%3A%2F%2Fwww%2East%2Eobs-mip%2Efr%2Fc-fica&urlhash=YN80&_t=tracking_disc
[http_sparse.ucd.ie_publications_rickard07duet.pdf]: http://www.linkedin.com/redirect?url=http%3A%2F%2Fsparse%2Eucd%2Eie%2Fpublications%2Frickard07duet%2Epdf&urlhash=fZ9d&_t=tracking_disc
[http_www.ast.obs-mip.fr_li-tifrom]: http://www.linkedin.com/redirect?url=http%3A%2F%2Fwww%2East%2Eobs-mip%2Efr%2Fli-tifrom&urlhash=PFff&_t=tracking_disc
[Randomized algorithms for matrices and data]: http://arxiv.org/abs/1104.5557
[Randomized Algorithms for Low-Rank Matrix Decomposition]: http://www.seas.upenn.edu/~bensapp/papers/wpe2-rand-low-rank.pdf
[Randomized PCA]: http://cims.nyu.edu/~tygert/software.html
[Blendenpik]: http://www.mathworks.com/matlabcentral/fileexchange/25241-blendenpik
[http_pdos.csail.mit.edu_petar_papers_blendenpik-v1.pdf]: http://pdos.csail.mit.edu/~petar/papers/blendenpik-v1.pdf
[RASL_ Robust Batch Alignment of Images by Sparse and Low-Rank Decomposition]: http://perception.csl.uiuc.edu/matrix-rank/rasl.html
[TILT_ Transform Invariant Low-rank Textures]: http://perception.csl.uiuc.edu/matrix-rank/tilt.html
[Scikit Learn]: http://scikit-learn.sourceforge.net/stable/modules/decomposition.html
[Matlab Toolbox for Dimensionality Reduction]: http://homepage.tudelft.nl/19j49/Matlab_Toolbox_for_Dimensionality_Reduction.html
[Orange]: http://orange.biolab.si/
[pcaMethods]: http://bioinformatics.oxfordjournals.org/content/23/9/1164.full
[bioconductor]: http://www.bioconductor.org/
[Danny Bickson]: http://www.linkedin.com/pub/danny-bickson/0/b32/835
[blog on GraphLab]: http://bickson.blogspot.com/
[Matrix Factorizations on Amazon]: http://astore.amazon.com/nuitblan-20?_encoding=UTF8&node=20
[CS_ Low Rank Compressive Spectral Imaging and a multishot CASSI]: http://nuit-blanche.blogspot.com/2011/01/cs-low-rank-compressive-spectral.html
[CS_ Heuristics for Rank Proxy and how it changes everything_.]: http://nuit-blanche.blogspot.com/2011/05/cs-heuristics-for-rank-proxy-and-how-it.html
[Welcome to the Matrix Factorization Jungle]: http://nuit-blanche.blogspot.com/2011/08/current-jungle-in-matrix-factorization.html
[Finding Structure With Randomness]: http://nuit-blanche.blogspot.com/2011/07/finding-structure-with-randomness.html
[http_blog.sciencenet.cn_blog-242887-483128.html]: http://blog.sciencenet.cn/blog-242887-483128.html