# A robust parallel algorithm for combinatorial compressed sensing

## TL;DR

This paper introduces the Robust-$\ell_0$ decoding algorithm, enhancing parallel combinatorial compressed sensing by effectively recovering sparse signals from noisy measurements, outperforming existing methods under Gaussian noise conditions.

## Contribution

The paper develops a robust decoding algorithm that extends previous work by incorporating noise resilience through posterior distribution approximation.

## Key findings

- Robust-$\ell_0$ outperforms existing algorithms in noisy environments.
- The method effectively recovers sparse signals with Gaussian noise.
- Analytic expressions for posteriors are derived under certain assumptions.

## 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 signal-to-noise ratios in the setting of Gaussian signals and Gaussian additive noise.