Compressed sensing with sparse binary matrices: Instance optimal error guarantees in near-optimal time

A compressed sensing method consists of a rectangular measurement matrix, M∈Rm×N with m≪N, together with an associated recovery algorithm, A:Rm→RN. Compressed sensing methods aim to construct a high quality approximation to any given input vector x∈RN using only Mx∈Rm as input. In particular, we foc...

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Veröffentlicht in:Journal of Complexity Jg. 30; H. 1; S. 1 - 15
1. Verfasser: Iwen, M.A.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier Inc 01.02.2014
Schlagworte:
Compressed sensing
Sparse matrices
Sublinear-time algorithms
Sparse matrices
Sublinear-time algorithms
Compressed sensing
ISSN:0885-064X, 1090-2708
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Zusammenfassung:A compressed sensing method consists of a rectangular measurement matrix, M∈Rm×N with m≪N, together with an associated recovery algorithm, A:Rm→RN. Compressed sensing methods aim to construct a high quality approximation to any given input vector x∈RN using only Mx∈Rm as input. In particular, we focus herein on instance optimal nonlinear approximation error bounds for M and A of the form ‖x−A(Mx)‖p≤‖x−xkopt‖p+Ck1/p−1/q‖x−xkopt‖q for x∈RN, where xkopt is the best possible k-term approximation to x. In this paper we develop a compressed sensing method whose associated recovery algorithm, A, runs in O((klogk)logN)-time, matching a lower bound up to a O(logk) factor. This runtime is obtained by using a new class of sparse binary compressed sensing matrices of near optimal size in combination with sublinear-time recovery techniques motivated by sketching algorithms for high-volume data streams. The new class of matrices is constructed by randomly subsampling rows from well-chosen incoherent matrix constructions which already have a sub-linear number of rows. As a consequence, fewer random bits than previously required are needed in order to select the rows utilized by the fast reconstruction algorithms considered herein.
ISSN:0885-064X
1090-2708
DOI:10.1016/j.jco.2013.08.001