Optimization Algorithms for Sparse Representations and Applications

We consider the following sparse representation problem, which is called Sparse Component Analysis: identify the matrices S ∈ IRn×N and A ∈ IRm×n (m ≤ n < N) uniquely (up to permutation of scaling), knowing only their multiplication X = AS, under some conditions, expressed either in terms of A and sparsity of S (identifiability conditions), or in terms of X (Sparse Component Analysis conditions). A crucial assumption (sparsity condition) is that S is sparse of level k in sense that each column of S has at most k nonzero elements (k = 1,2, ..., m − 1).

[1]  Terrence J. Sejnowski,et al.  Blind source separation of more sources than mixtures using overcomplete representations , 1999, IEEE Signal Processing Letters.

[2]  L. K. Hansen,et al.  Independent component analysis of functional MRI: what is signal and what is noise? , 2003, Current Opinion in Neurobiology.

[3]  J. Mazziotta,et al.  Rapid Automated Algorithm for Aligning and Reslicing PET Images , 1992, Journal of computer assisted tomography.

[4]  Barak A. Pearlmutter,et al.  Blind Source Separation by Sparse Decomposition in a Signal Dictionary , 2001, Neural Computation.

[5]  Andrzej Cichocki,et al.  Sparse component analysis of overcomplete mixtures by improved basis pursuit method , 2004, 2004 IEEE International Symposium on Circuits and Systems (IEEE Cat. No.04CH37512).

[6]  Michael Elad,et al.  Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[7]  Fathi M. Salem,et al.  ALGEBRAIC OVERCOMPLETE INDEPENDENT COMPONENT ANALYSIS , 2003 .

[8]  Axel Wismüller,et al.  Cluster Analysis of Biomedical Image Time-Series , 2002, International Journal of Computer Vision.

[9]  Fabian J. Theis,et al.  Blind source separation and sparse component analysis of overcomplete mixtures , 2004, 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[10]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[11]  E. Oja,et al.  Independent Component Analysis , 2001 .

[12]  Andrzej Cichocki,et al.  Adaptive blind signal and image processing , 2002 .

[13]  Fabian J. Theis,et al.  A geometric algorithm for overcomplete linear ICA , 2004, Neurocomputing.

[14]  Michael Zibulevsky,et al.  Underdetermined blind source separation using sparse representations , 2001, Signal Process..

[15]  Pando G. Georgiev,et al.  Optimization techniques for independent component analysis with applications to EEG data , 2004 .

[16]  Victoria Stodden,et al.  When Does Non-Negative Matrix Factorization Give a Correct Decomposition into Parts? , 2003, NIPS.

[17]  H. Sebastian Seung,et al.  Learning the parts of objects by non-negative matrix factorization , 1999, Nature.

[18]  Paul S. Bradley,et al.  k-Plane Clustering , 2000, J. Glob. Optim..

[19]  S Makeig,et al.  Analysis of fMRI data by blind separation into independent spatial components , 1998, Human brain mapping.