We introduce two new algorithms, Serial-ℓ0 and Parallel-ℓ0 for
solving a large underdetermined linear system of equations y=Ax∈Rm when it is known that x∈Rn 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-ℓ0 and Parallel-ℓ0 iteratively minimise ∥y−Ax^∥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 O(dnlogk) operations by assuming that the signal
is dissociated, meaning that all of the 2k 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-ℓ0
and Parallel-ℓ0 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∈Rn from m expander
measurements with n/m=103 and k/m up to four times greater than what is
achievable by ℓ1-regularization from dense Gaussian measurements.
Additionally, Serial-ℓ0 and Parallel-ℓ0 are observed to be able to
solve large problems sizes in substantially less time than other algorithms for
compressed sensing. In particular, Parallel-ℓ0 is structured to take
advantage of massively parallel architectures.Comment: 14 pages, 10 figure