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

• Authors: Massoud Babaie-Zadeh;Christian Jutten
• Venue: IEEE Transactions on Signal Processing
• Year: 2010

Let x be a signal to be sparsely decomposed over a redundant dictionary A, i.e., a sparse coefficient vectors has to be found such that x = A. 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∥2 ≤ δ rather than satisfying the exact equality x = As. Then, they have shown that if there is a decomposition with ∥s∥o 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∥o o o M-1)/2, because usually (1+M-1)/2 ≪ (1/2) spark(A). This limitation maybe had not been very important before, because ∥s∥o 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∥o M-1)/2 to the whole uniqueness range ∥s∥o