On the stable recovery of the sparsest overcomplete representations in presence of noise

Abstract

Let x be a signal to be sparsely decomposed over a redundant dictionary A, i.e., a sparse coefficient vector s has to be found such that x=As. It is known that this problem is inherently unstable against noise, and to overcome this instability, the authors of [Stable Recovery; Donoho et.al., 2006] have proposed to use an "approximate" decomposition, that is, a decomposition satisfying ||x - A s|| < \delta, rather than satisfying the exact equality x = As. Then, they have shown that if there is a decomposition with ||s||_0 < (1+M^{-1})/2, where M denotes the coherence of the dictionary, this decomposition would be stable against noise. On the other hand, it is known that a sparse decomposition with ||s||_0 < spark(A)/2 is unique. In other words, although a decomposition with ||s||_0 < spark(A)/2 is unique, its stability against noise has been proved only for highly more restrictive decompositions satisfying ||s||_0 < (1+M^{-1})/2, because usually (1+M^{-1})/2 << spark(A)/2. This limitation maybe had not been very important before, because ||s||_0 < (1+M^{-1})/2 is also the bound which guaranties that the sparse decomposition can be found via minimizing the L1 norm, a classic approach for sparse decomposition. However, with the availability of new algorithms for sparse decomposition, namely SL0 and Robust-SL0, it would be important to know whether or not unique sparse decompositions with (1+M^{-1})/2 < ||s||_0 < spark(A)/2 are stable. In this paper, we show that such decompositions are indeed stable. In other words, we extend the stability bound from ||s||_0 < (1+M^{-1})/2 to the whole uniqueness range ||s||_0 < spark(A)/2. In summary, we show that "all unique sparse decompositions are stably recoverable". Moreover, we see that sparser decompositions are "more stable".Comment: Accepted in IEEE Trans on SP on 4 May 2010. (c) 2010 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other work