Expander $\ell_0$Decoding
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 compressedsensing (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 informationpreserving 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2015
 arXiv:
 arXiv:1508.01256
 Bibcode:
 2015arXiv150801256M
 Keywords:

 Mathematics  Numerical Analysis;
 Computer Science  Information Theory;
 Mathematics  Combinatorics
 EPrint:
 14 pages, 10 figures