期刊
IEEE TRANSACTIONS ON INFORMATION THEORY
卷 55, 期 11, 页码 5302-5316出版社
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TIT.2009.2030471
关键词
Block restricted isometry property; block sparsity; compressed sensing; mixed-norm recovery; multiple measurement vectors (MMV); union of linear subspaces
资金
- Israel Science Foundation [1081/07]
- European Commission [216715]
Traditional sampling theories consider the problem of reconstructing an unknown signal from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which lies in a union of subspaces. In this paper, we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which lies in a sum of k subspaces, chosen from a larger set of m possibilities. The samples are modeled as inner products with an arbitrary set of sampling functions. To derive an efficient and robust recovery algorithm, we show that our problem can be formulated as that of recovering a block-sparse vector whose nonzero elements appear in fixed blocks. We then propose a mixed l(2)/l(1) program for block sparse recovery. Our main result is an equivalence condition under which the proposed convex algorithm is guaranteed to recover the original signal. This result relies on the notion of block restricted isometry property (RIP), which is a generalization of the standard RIP used extensively in the context of compressed sensing. Based on RIP, we also prove stability of our approach in the presence of noise and modeling errors. A special case of our framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Adapting our results to this context leads to new MMV recovery methods as well as equivalence conditions under which the entire set can be determined efficiently.
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