Recovery of Independent Sparse Sources From Linear Mixtures Using Sparse Bayesian Learning

Classical algorithms for the multiple measurement vector (MMV) problem assume either independent columns for the solution matrix or certain models of correlation among the columns. The correlation structure in the previous MMV formulation does not capture the signals well for some applications like photoplethysmography (PPG) signal extraction where the signals are independent and linearly mixed in a certain manner. In practice, the mixtures of these signals are observed through different channels. In order to capture this structure, we decompose the solution matrix into multiplication of a sparse matrix composed of independent components, and a linear mixing matrix. We derive a new condition that guarantees a unique solution for this linear mixing MMV problem. The condition can be much less restrictive than the conditions for the typical MMV problem in previous works. We also propose two novel sparse Bayesian learning (SBL) algorithms, independent component analysis sparse Bayesian learning, (ICASBL), and fast independent component sparse Bayesian learning, which capture the linear mixture structure. Analysis of the global and local minima of the ICASBL cost function is also provided, and similar to the typical SBL cost function it is shown that the local minima are sparse and that the global minima have maximum sparsity. Experimental results show that the proposed algorithms outperform traditional approaches and can recover the signal with fewer number of measurements in the linear mixing MMV setting.