TY - JOUR
T1 - Strengthening hash families and compressive sensing
AU - Colbourn, Charles
AU - Horsley, Daniel
AU - Syrotiuk, Violet
N1 - Funding Information:
Thanks to Chris McLean, Peyman Nayeri, and Devon OʼBrien, for helpful discussions. This research is supported by ARC DP120103067 (C.J.C., D.H.) and DE120100040 (D.H.).
PY - 2012/10
Y1 - 2012/10
N2 - The deterministic construction of measurement matrices for compressive sensing is a challenging problem, for which a number of combinatorial techniques have been developed. One of them employs a widely used column replacement technique based on hash families. It is effective at producing larger measurement matrices from smaller ones, but it can only preserve the strength (level of sparsity supported), not increase it. Column replacement is extended here to produce measurement matrices with larger strength from ingredient arrays with smaller strength. To do this, a new type of hash family, called a strengthening hash family, is introduced. Using these hash families, column replacement is shown to increase strength under two standard notions of recoverability. Then techniques to construct strengthening hash families, both probabilistically and deterministically, are developed. Using a variant of the Stein-Lovász-Johnson theorem, a deterministic, polynomial time algorithm for constructing a strengthening hash family of fixed strength is derived.
AB - The deterministic construction of measurement matrices for compressive sensing is a challenging problem, for which a number of combinatorial techniques have been developed. One of them employs a widely used column replacement technique based on hash families. It is effective at producing larger measurement matrices from smaller ones, but it can only preserve the strength (level of sparsity supported), not increase it. Column replacement is extended here to produce measurement matrices with larger strength from ingredient arrays with smaller strength. To do this, a new type of hash family, called a strengthening hash family, is introduced. Using these hash families, column replacement is shown to increase strength under two standard notions of recoverability. Then techniques to construct strengthening hash families, both probabilistically and deterministically, are developed. Using a variant of the Stein-Lovász-Johnson theorem, a deterministic, polynomial time algorithm for constructing a strengthening hash family of fixed strength is derived.
KW - Column replacement
KW - Compressive sensing
KW - Hash family
KW - Lovász local lemma
KW - Stein-Lovász- Johnson theorem
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U2 - 10.1016/j.jda.2012.04.004
DO - 10.1016/j.jda.2012.04.004
M3 - Article
AN - SCOPUS:84865505930
SN - 1570-8667
VL - 16
SP - 170
EP - 186
JO - Journal of Discrete Algorithms
JF - Journal of Discrete Algorithms
ER -