On the Use of A Priori Information for Sparse Signal Representations

This report studies the effect of introducing a priori knowledge to recover sparse representations when overcomplete dictionaries are used. We focus mainly on Greedy algorithms and Basis Pursuit as for our algorithmic basement, while a priori is incorporated by suitably weighting the elements of the dictionary. A unique sufficient condition is provided under which Orthogonal Matching Pursuit, Matching Pursuit and Basis Pursuit are able to recover the optimally sparse representation of a signal when a priori information is available. Theoretical results show how the use of “reliable” a priori information can improve the performances of these algorithms. In particular, we prove that sufficient conditions to guarantee the retrieval of the sparsest solution can be established for dictionaries unable to satisfy the results of Gribonval and Vandergheynst [1] and Tropp [2]. As one might expect, our results reduce to the classical case of [1] and [2] when no a priori information is available. Some examples illustrate our theoretical findings. Index Terms Sparse Representations, Basis Pursuit, Matching Pursuit, Greedy Algorithms, A Priori Knowledge.