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• W2161386354 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 $|y - Ahat x|_0$ 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 {em 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 $xinmathbb{R}^n$ 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, due to their low computational complexity, 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." @default.
• W2161386354 created "2016-06-24" @default.
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• W2161386354 date "2015-08-06" @default.
• W2161386354 modified "2023-09-27" @default.
• W2161386354 title "Expander $ell_0$-Decoding" @default.
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