Sparse recovery with partial support knowledge
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Indyk_Sparse recovery with parital support.pdf
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Author(s)
Do Ba, Khanh
Indyk, Piotr
Date Issued
August 2011
Journal
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Publisher
Springer Berlin / Heidelberg
Citation
Ba, Khanh Do, and Piotr Indyk. “Sparse Recovery with Partial Support Knowledge.” Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. Ed. Leslie Ann Goldberg et al. LNCS Vol. 6845. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. 26–37.
Version
Author's final manuscript
Abstract
The goal of sparse recovery is to recover the (approximately) best k-sparse approximation [ˆ over x] of an n-dimensional vector x from linear measurements Ax of x. We consider a variant of the problem which takes into account partial knowledge about the signal. In particular, we focus on the scenario where, after the measurements are taken, we are given a set S of size s that is supposed to contain most of the “large” coefficients of x. The goal is then to find [ˆ over x] such that [ ||x-[ˆ over x]|| [subscript p] ≤ C min ||x-x'||[subscript q]. [over] k-sparse x' [over] supp (x') [c over _] S]
We refer to this formulation as the sparse recovery with partial support knowledge problem ( SRPSK ). We show that SRPSK can be solved, up to an approximation factor of C = 1 + ε, using O( (k/ε) log(s/k)) measurements, for p = q = 2. Moreover, this bound is tight as long as s = O(εn / log(n/ε)). This completely resolves the asymptotic measurement complexity of the problem except for a very small range of the parameter s.
To the best of our knowledge, this is the first variant of (1 + ε)-approximate sparse recovery for which the asymptotic measurement complexity has been determined.
Description
14th International Workshop, APPROX 2011, and 15th International Workshop, RANDOM 2011, Princeton, NJ, USA, August 17-19, 2011. Proceedings
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1007/978-3-642-22935-0_3
