A hierarchical approach for sparse source Blind Signal Separation problem

In this paper, a Blind Signal Separation (BSS) problem is considered: given [email protected]?R^m^x^N, BSS problem is to find [email protected]?R^m^x^n and [email protected]?R^n^x^N, where the matrices are related as X=AS. We have reviewed the sufficient conditions on the structure of X, A and S in terms of sparseness conditions on S, such that the equation X=AS can be solved uniquely (up to permutation and scalability). A hierarchical 0-1 MIP is proposed to solve the problem. Probabilistically, we have shown that every subsequent level of hierarchical MIP will be easier to solve than the precedent level of MIP. Moreover, we have presented case studies that illustrate the performance of proposed solution approach for correlated sparse sources.

[1]  Fabian J Theis,et al.  Latent Variable Analysis and Signal Separation : 10th International Conference, LVA/ICA 2012, Tel Aviv, Israel, March 12-15, 2012. Proceedings , 2012 .

[2]  Pat Morin,et al.  Covering Things with Things , 2005, Discret. Comput. Geom..

[3]  Guillaume Gelle,et al.  BLIND SOURCE SEPARATION: A TOOL FOR ROTATING MACHINE MONITORING BY VIBRATIONS ANALYSIS? , 2001 .

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

[5]  Fabian J. Theis,et al.  Sparse component analysis and blind source separation of underdetermined mixtures , 2005, IEEE Transactions on Neural Networks.

[6]  Erkki Oja,et al.  Independent component analysis: algorithms and applications , 2000, Neural Networks.

[7]  Hiroshi Konno,et al.  A cutting plane algorithm for solving bilinear programs , 1976, Math. Program..

[8]  Edoardo Amaldi,et al.  The MIN PFS problem and piecewise linear model estimation , 2002, Discret. Appl. Math..

[9]  Panos M. Pardalos,et al.  Introduction to Global Optimization , 2000, Introduction to Global Optimization.

[10]  Michael Zibulevsky,et al.  Sparse Component Analysis , 2010 .

[11]  Panos M. Pardalos,et al.  A bilinear algorithm for sparse representations , 2007, Comput. Optim. Appl..

[12]  Joseph F. Murray,et al.  Dictionary Learning Algorithms for Sparse Representation , 2003, Neural Computation.

[13]  Reiner Horst,et al.  Introduction to Global Optimization (Nonconvex Optimization and Its Applications) , 2002 .

[14]  Fabian J. Theis,et al.  Blind Source Separation of Linear Mixtures with Singular Matrices , 2004, ICA.

[15]  Warren P. Adams,et al.  A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems , 1998 .

[16]  Karin Schnass,et al.  Dictionary Identification—Sparse Matrix-Factorization via $\ell_1$ -Minimization , 2009, IEEE Transactions on Information Theory.

[17]  Fabian J. Theis,et al.  Sparse Component Analysis: a New Tool for Data Mining , 2004 .

[18]  Ekrem Gursoy,et al.  Independent Component Analysis Techniques for Power System Load Estimation , 2005 .

[19]  E. Oja,et al.  Independent Component Analysis , 2013 .

[20]  Asoke K. Nandi,et al.  Blind Source Separation , 1999 .

[21]  Pankaj K. Agarwal,et al.  Approximation Algorithms for k-Line Center , 2002, ESA.

[22]  Andrzej Cichocki,et al.  New Algorithms for Non-Negative Matrix Factorization in Applications to Blind Source Separation , 2006, 2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings.

[23]  Panos M. Pardalos,et al.  Multilevel (Hierarchical) Optimization: Complexity Issues, Optimality Conditions, Algorithms , 2009 .

[24]  Pankaj K. Agarwal,et al.  Approximation algorithms for projective clustering , 2000, SODA '00.

[25]  Emmanuel Vincent,et al.  Proceedings of the 10th international conference on Latent Variable Analysis and Signal Separation , 2010 .

[26]  Zvi Drezner,et al.  Facility location - applications and theory , 2001 .

[27]  A. Bruckstein,et al.  On the uniqueness of overcomplete dictionaries, and a practical way to retrieve them , 2006 .

[28]  Te-Won Lee,et al.  Independent Component Analysis , 1998, Springer US.

[29]  Christian Jutten,et al.  Detection de grandeurs primitives dans un message composite par une architecture de calcul neuromime , 1985 .

[30]  Pierre Comon,et al.  Handbook of Blind Source Separation: Independent Component Analysis and Applications , 2010 .

[31]  I Daubechies,et al.  Independent component analysis for brain fMRI does not select for independence , 2009 .

[32]  Kanika Dhyani,et al.  Optimization models and algorithms for the hyperplane clustering problem , 2010, 4OR.

[33]  Panos M. Pardalos,et al.  Multilevel Optimization: Algorithms and Applications , 2012 .