On the Stable Recovery of the Sparsest Overcomplete Representations in Presence of Noise
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 ["Stable recovery of sparse overcomplete representations in the presence of noise," IEEE Trans. Inf. Theory, vol. 52, no. 1, pp. 6-18, Jan. 2006] have proposed to use an "approximate" decomposition, that is, a decomposition satisfying ∥x - As∥<sub>2</sub> ≤ δ rather than satisfying the exact equality x = As. Then, they have shown that if there is a decomposition with ∥s∥<sub>0</sub> <;; (1 + M<sup>-1</sup>)/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∥<sub>0</sub> <;; (1/2)spark(A) is unique. In other words, although a decomposition with ∥s∥<sub>0</sub> <;; (1/2)spark(A) is unique, its stability against noise has been proved only for highly more restrictive decompositions satisfying ∥s∥<sub>0</sub> <;; (1 + M<sup>-1</sup>)/2, because usually (1 + M<sup>-1</sup>)/2 ≪ (1/2)spark(A). This limitation maybe had not been very important before, because ∥s∥<sub>0</sub> <;; (1 + M<sup>-1</sup>)/2 is also the bound which guaranties that the sparse decomposition can be found via minimizing the l<sup>1</sup> norm, a classic approach for sparse decomposition. However, with the availability of new algorithms for sparse decomposition, namely SL0 and robust-SLO, it would be important to know whether or not unique sparse decompositions with (1 + M<sup>-1</sup>)/2 ≤ ∥s∥<sub>0</sub> <;; (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 ∥s∥<sub>0</sub> <;; (1 + M<sup>-1</sup>)/2 to the whole uniqueness range ∥s∥0 <;; (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".