Conditioning of Random Block Subdictionaries with Applications to Block-Sparse Recovery and Regression

This paper makes several contributions toward the "underdetermined" setting in linear models when the set of observations is given by a linear combination of a small number of groups of columns of a dictionary, termed the "block-sparse" case. First, it specifies conditions on the dictionary under which most block submatrices of the dictionary are well conditioned. In contrast to earlier works in block-sparse inference, this result is fundamentally different because (i) it provides conditions that can be explicitly computed in polynomial time, (ii) the provided conditions translate into near-optimal scaling of the number of columns of the block subdictionaries as a function of the number of observations for a large class of dictionaries, and (iii) it suggests that the spectral norm, rather than the column/block coherences, of the dictionary fundamentally limits the dimensions of the well-conditioned block subdictionaries. Second, in order to help understand the significance of this result in the context of block-sparse inference, this paper investigates the problems of block-sparse recovery and block-sparse regression in underdetermined settings. In both of these problems, this paper utilizes its result concerning conditioning of block subdictionaries and establishes that near-optimal block-sparse recovery and block-sparse regression is possible for a large class of dictionaries as long as the dictionary satisfies easily computable conditions and the coefficients describing the linear combination of groups of columns can be modeled through a mild statistical prior. Third, the paper reports carefully constructed numerical experiments that highlight the effects of different measures of the dictionary in block-sparse inference problems.