A Counter-Example to the General Convergence of Partially Greedy Algorithms

Abstract

In a separable Hilbert space H, greedy algorithms iteratively define m-term approximants to a given vector from a complete redundant dictionary D. With very large dictionaries, the pure greedy algorithm cannot be implemented and must be replaced with a weak greedy algorithm. In numerical applications, partially greedy algorithms have been introduced to reduce the numerical complexity. A conjecture about their convergence arises naturally from the observation of numerical experiments. We introduce, study and disprove this conjecture.