Near Optimal Compressed Sensing of Sparse Rank-One Matrices via Sparse Power Factorization

Compressed sensing of simultaneously sparse and rank-one matrices enables recovery of sparse signals from a few linear measurements of their bilinear form. One important question is how many measurements are needed for a stable reconstruction in the presence of measurement noise. Unlike the conventional compressed sensing for sparse vectors, where convex relaxation via the $\ell_1$-norm achieves near optimal performance, for compressed sensing of sparse and rank-one matrices, recently it has been shown by Oymak et al. that convex programmings using the nuclear norm and the mixed norm are highly suboptimal even in the noise-free scenario. We propose an alternating minimization algorithm called sparse power factorization (SPF) for compressed sensing of sparse rank-one matrices. Starting from a particular initialization, SPF achieves stable recovery and requires number of measurements within a logarithmic factor of the information-theoretic fundamental limit. For fast-decaying sparse signals, SPF starting from an initialization with low computational cost also achieves stable reconstruction with the same number of measurements. Numerical results show that SPF empirically outperforms the best known combinations of mixed norm and nuclear norm.