Sparse Representation and Its Applications in Blind Source Separation

Abstract

In this paper, sparse representation (factorization) of a data matrix is first discussed. An overcomplete basis matrix is estimated by using the K(cid:0)means method. We have proved that for the estimated overcom- plete basis matrix, the sparse solution (coefficient matrix) with minimum l1(cid:0)norm is unique with probability of one, which can be obtained using a linear programming algorithm. The comparisons of the l1(cid:0)norm so- lution and the l0(cid:0)norm solution are also presented, which can be used in recoverability analysis of blind source separation (BSS). Next, we ap- ply the sparse matrix factorization approach to BSS in the overcomplete case. Generally, if the sources are not sufficiently sparse, we perform blind separation in the time-frequency domain after preprocessing the observed data using the wavelet packets transformation. Third, an EEG experimental data analysis example is presented to illustrate the useful- ness of the proposed approach and demonstrate its performance. Two almost independent components obtained by the sparse representation method are selected for phase synchronization analysis, and their peri- ods of significant phase synchronization are found which are related to tasks. Finally, concluding remarks review the approach and state areas that require further study.