Sketching sparse low-rank matrices with near-optimal sample- and time-complexity
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We consider the problem of recovering an n_1 × 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. 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 numerical simulations.
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