Average-Case Analysis of Greedy Pursuit

Recent work on sparse approximation has focused on the theoretical performance of algorithms for random inputs. This average-case behavior is typically far better than the behavior of the algorithm for the worst inputs. Moreover, an average-case analysis fits naturally with the type of signals that arise in certain applications, such as wireless communications. This paper describes what is currently known about the performance of greedy pursuit algorithms with random inputs. In particular, it gives a new result for the performance of Orthogonal Matching Pursuit (OMP) for sparse signals contaminated with random noise, and it explains recent work on recovering sparse signals from random measurements via OMP. The paper also provides a list of open problems to stimulate further research.