Expander $\ell_0$-Decoding

TL;DR

This paper introduces two efficient algorithms, Serial-$ ext{ extlbrackdbl} ext{ extlbrackdbl} ext{ extlbrackdbl}$ and Parallel-$ ext{ extlbrackdbl} ext{ extlbrackdbl} ext{ extlbrackdbl}$, for sparse signal recovery using expander graphs, outperforming traditional methods in large-scale, low-measurement compressed sensing tasks.

Contribution

The paper presents novel Serial-$ ext{ extlbrackdbl} ext{ extlbrackdbl} ext{ extlbrackdbl}$ and Parallel-$ ext{ extlbrackdbl} ext{ extlbrackdbl} ext{ extlbrackdbl}$ algorithms that leverage expander graph properties for efficient sparse recovery, with proven convergence and superior empirical performance.

Findings

Able to recover larger sparsity levels than $ ext{ extlbrackdbl}$-regular $ ext{ extlbrackdbl}$ algorithms.

Achieves recovery with fewer measurements compared to $ ext{ extlbrackdbl}$-regular algorithms.

Runs significantly faster on large-scale problems, especially with parallel architecture.

Abstract

We introduce two new algorithms, Serial-$\ell_0$ and Parallel-$\ell_0$ for solving a large underdetermined linear system of equations $y = Ax \in \mathbb{R}^m$ when it is known that $x \in \mathbb{R}^n$ has at most $k < m$ nonzero entries and that $A$ is the adjacency matrix of an unbalanced left $d$-regular expander graph. The matrices in this class are sparse and allow a highly efficient implementation. A number of algorithms have been designed to work exclusively under this setting, composing the branch of combinatorial compressed-sensing (CCS). Serial-$\ell_0$ and Parallel-$\ell_0$ iteratively minimise $\|y - A\hat x\|_0$ by successfully combining two desirable features of previous CCS algorithms: the information-preserving strategy of ER, and the parallel updating mechanism of SMP. We are able to link these elements and guarantee convergence in $\mathcal{O}(dn \log k)$ operations by assuming that the signal is dissociated, meaning that all of the $2^k$ subset sums of the support of $x$ are pairwise different. However, we observe empirically that the signal need not be exactly dissociated in practice. Moreover, we observe Serial-$\ell_0$ and Parallel-$\ell_0$ to be able to solve large scale problems with a larger fraction of nonzeros than other algorithms when the number of measurements is substantially less than the signal length; in particular, they are able to reliably solve for a $k$-sparse vector $x\in\mathbb{R}^n$ from $m$ expander measurements with $n/m=10^3$ and $k/m$ up to four times greater than what is achievable by $\ell_1$-regularization from dense Gaussian measurements. Additionally, Serial-$\ell_0$ and Parallel-$\ell_0$ are observed to be able to solve large problems sizes in substantially less time than other algorithms for compressed sensing. In particular, Parallel-$\ell_0$ is structured to take advantage of massively parallel architectures.