A Robust Parallel Algorithm for Combinatorial Compressed Sensing
Abstract
In previous work two of the authors have shown that a vector $x \in \mathbb{R}^n$ with at most $k < n$ nonzeros can be recovered from an expander sketch $Ax$ in $\mathcal{O}(\mathrm{nnz}(A)\log k)$ operations via the Parallel$\ell_0$ decoding algorithm, where $\mathrm{nnz}(A)$ denotes the number of nonzero entries in $A \in \mathbb{R}^{m \times n}$. In this paper we present the Robust$\ell_0$ decoding algorithm, which robustifies Parallel$\ell_0$ 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$\ell_0$ is superior to existing greedy and combinatorial compressed sensing algorithms in the presence of small to moderate signaltonoise ratios in the setting of Gaussian signals and Gaussian additive noise.
 Publication:

IEEE Transactions on Signal Processing
 Pub Date:
 April 2018
 DOI:
 10.1109/TSP.2018.2806359
 arXiv:
 arXiv:1704.09012
 Bibcode:
 2018ITSP...66.2167M
 Keywords:

 Mathematics  Numerical Analysis;
 Computer Science  Information Theory
 EPrint:
 doi:10.1109/TSP.2018.2806359