Sharp Convergence Rates for Matching Pursuit

We study the fundamental limits of matching pursuit, or the pure greedy algorithm, for approximating a target function by a sparse linear combination of elements from a dictionary. When the target function is contained in the variation space corresponding to the dictionary, many impressive works over the past few decades have obtained upper and lower bounds on the convergence rate of matching pursuit, but they do not match. The main contribution of this paper is to close this gap and obtain a sharp characterization of the performance of matching pursuit. We accomplish this by improving the existing lower bounds to match the best upper bound. Specifically, we construct a worst case dictionary which proves that the existing upper bound cannot be improved. It turns out that, unlike other greedy algorithm variants, the converge rate is suboptimal and is determined by the solution to a certain non-linear equation. This enables us to conclude that any amount of shrinkage improves matching pursuit in the worst case.