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 [1]. 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.

The sources for this list include the following most excellent sites: 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 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 or. 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
• ** 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.” by MichaleI Mccoy and Joel Tropp
• SPAMS (SPArse Modeling Software)
• ADMM: Alternating Direction Method of Multipliers ‘‘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 - MATLABzip], 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] [partial SVD version - MATLAB zip]
• Dual Method [MATLAB zip], 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]. Reference - A Singular Value Thresholding Algorithm for Matrix Completion, J. -F. Cai, E. J. Candès, and Z. Shen (2008).
• Alternating Direction Method [MATLAB zip] , Reference - Sparse and Low-Rank Matrix Decomposition via Alternating Direction Methods, X. Yuan, and J. Yang (2009).
• 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
• DSPCA: Sparse PCA using SDP . Code is here.
• PathPCA: A fast greedy algorithm for Sparse PCA. The code is here.

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 Softwareor SPAMS]
• Dictionary Learning Algorithms for Sparse Representation (Matlab implementation of 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 ]
• Efficient sparse coding algorithms [ Matlab code is here ]
• Shift Invariant Sparse Coding of Image and Music Data. Matlab implemention is here
• Shift-invariant dictionary learning for sparse representations: extending K-SVD.
• 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) [Matlab implementation of NMF (Non-negative Matrix Factorization) and NTF (Non-negative Tensor), a faster implementation of NMF can be found here, 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

• HALS: Accelerated Multiplicative Updates and Hierarchical ALS Algorithms for Nonnegative Matrix Factorization by Nicolas Gillis, Fran?ois Glineur.
• 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 contains an optimized C implementation of the Non-Negative Matrix Factorization (NMF) algorithm, described in [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].
• NTFLAB for Signal Processing, 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) [Matlab implementation of NMF (Non-negative Matrix Factorization) and NTF (Non-negative Tensor), a faster implementation of NMF can be found here, 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 by Zhilin Zhang
• Compressive MUSIC with optimized partial support for joint sparse recovery by Jong Min Kim, Ok Kyun Lee, Jong Chul Ye [no code]
• 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/
• BLISS softwares: http://www.lis.inpg.fr/pages_perso/bliss/deliverables/d20.html
• MISEP: 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
• C-FICA: http://www.ast.obs-mip.fr/c-fica

SCA:

• DUET: 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

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 )

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
• 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: Heuristics for Rank Proxy and how it changes everything….
• Tennis Players are Sparse !

Sources

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

• Raghunandan H. Keshavan‘ s list
• Stephen Becker’s list
• 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 Uni?ed View of Matrix Factorization Models by Ajit P. Singh and Geoffrey J. Gordon

Matrix Factorization, Algorithms, Applications, and Avaliable packages