Matrix Factorization, Algorithms, Applications, and Avaliable packages

来源：http://www.cvchina.info/2011/09/05/matrix-factorization-jungle/

美帝的有心人士收集了市面上的矩阵分解的差点儿全部算法和应用。因为源地址在某神奇物质之外，特转载过来，源地址

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 thelog-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 andJoel 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 ishere.
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 Vectorsby 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

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