Why Gabor Frames? Two Fundamental Measures of Coherence and their Geometric Significance

In the standard compressed sensing paradigm, the N x C measurement matrix is required to act as a near isometry on all k-sparse signals. This is the Restricted Isometry Property or k-RIP. It is known that certain probabilistic processes generate measurement or sensing matrices that satisfy k-RIP with high probability. However, no polynomial-time algorithm is known for verifying that a sensing matrix with the worst-case coherence \mu satisfies k-RIP with k greater than \mu^{-1}. In contrast, this paper provides simple conditions that, when satisfied, guarantee that a deterministic sensing matrix acts as a near isometry on all but an exponentially small fraction of k-sparse signals. These conditions are defined in terms of the worst-case coherence \mu and the expected coherence \nu among the columns of the measurement matrix. Under the assumption that C >= N^2 and \nu <= N^{-1}, the sparsity level k is determined by \mu^{-2}, while the fraction of "bad" k-sparse signals is determined by \nu^{-2} and \mu^{-2}. In contrast to the k-RIP condition, these conditions are also extremely easy to check. Applying these conditions to Gabor frames shows that it is possible to successfully recover k-sparse signals for k=O(\mu^{-2}). In particular, this implies that Gabor frames generated from the Alltop sequence can successfully recover all but an exponentially small fraction of $k$-sparse signals for k=O(N).