A note on compressed sensing and the complexity of matrix multiplication

Authors:
M. A. Iwen;C. V. Spencer
Affiliations:
Institute for Mathematics and its Applications, University of Minnesota, 400 Lind Hall, 207 Church Street S.E., Minneapolis, MN 55455, USA;Institute for Advanced Study, School of Mathematics, 1 Einstein Drive, Princeton, NJ 08540, USA
Venue:
Information Processing Letters
Year:
2009

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Abstract

We consider the conjectured O(N^2^+^@e) time complexity of multiplying any two NxN matrices A and B. Our main result is a deterministic Compressed Sensing (CS) algorithm that both rapidly and accurately computes A@?B provided that the resulting matrix product is sparse/compressible. As a consequence of our main result we increase the class of matrices A, for any given NxN matrix B, which allows the exact computation of A@?B to be carried out using the conjectured O(N^2^+^@e) operations. Additionally, in the process of developing our matrix multiplication procedure, we present a modified version of Indyk's recently proposed extractor-based CS algorithm [P. Indyk, Explicit constructions for compressed sensing of sparse signals, in: SODA, 2008] which is resilient to noise.