Blind source separation and sparse component analysis of overcomplete mixtures

We formulate conditions (k-SCA-conditions) under which we can represent a given (m x N)-matrix X (data set) uniquely (up to scaling and permutation) as a multiplication of m x n and n x N matrices A and S (often called mixing matrix or dictionary and source matrix, respectively), such that S is sparse of level n-m+k in sense that each column of S has at least n - m + k zero elements. We call this the k-Sparse Component Analysis problem (k-SCA). Conditions on a matrix S are presented such that the k-SCA-conditions are satisfied for the matrix X = AS, where A is an arbitrary matrix from some class. This is the Blind Source Separation problem and the above conditions are called identifiability conditions. We present new algorithms: for matrix identification (under k-SCA-conditions), and for source recovery (under identifiability conditions). The methods are illustrated with examples, showing good separation of the high-frequency part of mixtures of images after appropriate sparsification.