The Statistical Coherence-based Theory of Robust Recovery of Sparsest Overcomplete Representation

Abstract

The recovery of sparsest overcomplete representation has recently attracted intensive research activities owe to its important potential in the many applied fields such as signal processing, medical imaging, communication, and so on. This problem can be stated in the following, i.e., to seek for the sparse coefficient vector of the given noisy observation over a redundant dictionary such that, where is the corrupted error. Elad et al. made the worst-case result, which shows the condition of stable recovery of sparest overcomplete representation over is where . Although it's of easy operation for any given matrix, this result can't provide us realistic guide in many cases. On the other hand, most of popular analysis on the sparse reconstruction relies heavily on the so-called RIP (Restricted Isometric Property) for matrices developed by Candes et al., which is usually very difficult or impossible to be justified for a given measurement matrix. In this article, we introduced a simple and efficient way of determining the ability of given D used to recover the sparse signal based on the statistical analysis of coherence coefficients, where is the coherence coefficients between any two different columns of given measurement matrix . The key mechanism behind proposed paradigm is the analysis of statistical distribution (the mean and covariance) of . We proved that if the resulting mean of are zero, and their covariance are as small as possible, one can faithfully recover approximately sparse signals from a minimal number of noisy measurements with overwhelming probability. The resulting theory is not only suitable for almost all models - e.g. Gaussian, frequency measurements-discussed in the literature of compressed sampling, but also provides a framework for new measurement strategies as well.