Expander $\ell_0$-Decoding

We introduce two new algorithms, Serial-`0 and Parallel-`0 for solving a large underdetermined linear system of equations y = Ax ∈ R when it is known that x ∈ R 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-`0 and Parallel-`0 iteratively minimise ‖y − Ax̂‖0 by successfully combining two desirable features of previous CCS algorithms: the information-preserving strategy of ER [1], and the parallel updating mechanism of SMP [2]. We are able to link these elements and guarantee convergence in O(dn log k) operations by assuming that the signal is dissociated, meaning that all of the 2 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-`0 and Parallel-`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 ∈ R from m expander measurements with n/m = 10 and k/m up to four times greater than what is achievable by `1regularization from dense Gaussian measurements. Additionally, due to their low computational complexity, Serial-`0 and Parallel`0 are observed to be able to solve large problems sizes in substantially less time than other algorithms for compressed sensing. In particular, Parallel-`0 is structured to take advantage of massively parallel architectures.