On the Stable Recovery of the Sparsest Overcomplete Representations in Presence of Noise
Résumé
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, Donoho, Elad and Temlyakov (IEEE Trans. IT, Jan. 2006) have proposed to use an “approximate” decomposition, that is, a decomposition such that the square norm of x - As is less than a small positive real rather than satisfying the exact equality x = As. Then, they have shown that if there is a decomposition with a L0-norm less than (1+Mi)/2 , where Mi denotes the inverse of the coherence of the dictionary, this decomposition would be stable against noise. On the other hand, it is known that a sparse decomposition with L0-norm of s les than (1/2)spark(A) is unique. In other words, although the former decomposition is unique, its stability against noise has been proved only for highly more restrictive decompositions satisfying L0-norm of s less than (1+Mi)/2. This limitation maybe had not been very important before, because (1+Mi)/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 L0-norm in the range [(1+Mi)/2, (1/2)spark(A)] are stable. In this correspondence, we show that such decompositions are indeed stable. In other words, we extend the stability bound from L0-norm of s less than (1+Mi)/2 to the whole uniqueness range: L0-norm of s less than (1/2)spark(A). In summary, we show that all unique sparse decompositions are stably recoverable. Moreover, we see that sparser decompositions are “more stable.”
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