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Optimally sparse representation in general (non-orthogonal) dictionaries via ℓ¹ minimization (2002)
| Venue: | PROC. NATL ACAD. SCI. USA 100 2197–202 |
| Citations: | 630 - 38 self |
BibTeX
@INPROCEEDINGS{Donoho02optimallysparse,
author = {David L. Donoho and Michael Elad},
title = { Optimally sparse representation in general (non-orthogonal) dictionaries via ℓ¹ minimization},
booktitle = {PROC. NATL ACAD. SCI. USA 100 2197–202},
year = {2002},
publisher = {}
}
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Abstract
Given a ‘dictionary’ D = {dk} of vectors dk, we seek to represent a signal S as a linear combination S = ∑ k γ(k)dk, with scalar coefficients γ(k). In particular, we aim for the sparsest representation possible. In general, this requires a combinatorial optimization process. Previous work considered the special case where D is an overcomplete system consisting of exactly two orthobases, and has shown that, under a condition of mutual incoherence of the two bases, and assuming that S has a sufficiently sparse representation, this representation is unique and can be found by solving a convex optimization problem: specifically, minimizing the ℓ¹ norm of the coefficients γ. In this paper, we obtain parallel results in a more general setting, where the dictionary D can arise from two or several bases, frames, or even less structured systems. We introduce the Spark, ameasure of linear dependence in such a system; it is the size of the smallest linearly dependent subset (dk). We show that, when the signal S has a representation using less than Spark(D)/2 nonzeros, this representation is necessarily unique. We
Keyphrases
sparse representation combinatorial optimization process mutual incoherence scalar coefficient linear dependence structured system parallel result linear combination overcomplete system several base general setting convex optimization problem previous work linearly dependent subset special case dictionary dk
