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arXiv · 0904.0068

Verifiable conditions of $\ell_1$-recovery of sparse signals with sign restrictions

Anatoli Iouditski·Fatma Kilinc Karzan·Arkadii S. Nemirovski
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Abstract

We propose necessary and sufficient conditions for a sensing matrix to be "s-semigood" -- to allow for exact $\ell_1$-recovery of sparse signals with at most $s$ nonzero entries under sign restrictions on part of the entries. We express the error bounds for imperfect $\ell_1$-recovery in terms of the characteristics underlying these conditions. Furthermore, we demonstrate that these characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse $\ell_1$-recovery and to efficiently computable upper bounds on those $s$ for which a given sensing matrix is $s$-semigood. We concentrate on the properties of proposed verifiable sufficient conditions of $s$-semigoodness and describe their limits of performance.

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Anatoli Iouditski, Fatma Kilinc Karzan, Arkadii S. Nemirovski. 2010-05-31. Verifiable conditions of $\ell_1$-recovery of sparse signals with sign restrictions. https://doi.org/10.1007/s10107-010-0418-y

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