A robust parallel algorithm for combinatorial compressed sensing

It was shown in [1] that a vector x ∈ R n with at most k < n nonzeros can be recovered from an expander sketch Ax in O(nnz(A) log k) operations via the Parallel-`0 decoding algorithm, where nnz(A) denotes the number of nonzero entries in A ∈ R m×n . In this paper we present the Robust-`0 decoding algorithm, which robustifies Parallel-`0 when the sketch Ax is corrupted by additive noise. This robustness is achieved by approximating the asymptotic posterior distribution of values in the sketch given its corrupted measurements. We provide analytic expressions that approximate these posteriors under the assumptions that the nonzero entries in the signal and the noise are drawn from continuous distributions. Numerical experiments presented show that Robust-`0 is superior to existing greedy and combinatorial compressed sensing algorithms in the presence of small to moderate signal-to-noise ratios in the setting of Gaussian signals and Gaussian additive noise.