Sketching Sparse Low-Rank Matrices With Near-Optimal Sample- and Time-Complexity Using Message Passing

Abstract

We consider the problem of recovering an $n_{1} \times n_{2}$ low-rank matrix with $k$ -sparse singular vectors from a small number of linear measurements (sketch). We propose a sketching scheme and an algorithm that can recover the singular vectors with high probability, with a sample complexity and running time that both depend only on $k$ and not on the ambient dimensions $n_{1}$ and $n_{2}$ . Our sketching operator, based on a scheme for compressed sensing by Li et al. and Bakshi et al., uses a combination of a sparse parity check matrix and a partial DFT matrix. Our main contribution is the design and analysis of a two-stage iterative algorithm which recovers the singular vectors by exploiting the simultaneously sparse and low-rank structure of the matrix. We derive a nonasymptotic bound on the probability of exact recovery, which holds for any $n_{1}\times n_{2} $ sparse, low-rank matrix. We also show how the scheme can be adapted to tackle matrices that are approximately sparse and low-rank. The theoretical results are validated by extensive numerical simulations and comparisons with existing schemes that use convex optimization for recovery.