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 error 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 decay rate of matching pursuit. Specifically, we construct a worst case dictionary which shows that the existing best upper bound cannot be significantly 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.

翻译：本文研究匹配追踪（即纯贪婪算法）在通过字典元素的稀疏线性组合逼近目标函数时的基本极限。当目标函数位于字典对应的变差空间中时，过去数十年的诸多重要工作已获得匹配追踪误差的上界与下界，但两者并不匹配。本文的主要贡献在于填补这一差距，并给出匹配追踪衰减率的精确刻画。具体而言，我们构造了一个最坏情况字典，证明现有最优上界无法显著改进。结果表明，与其他贪婪算法变体不同，其收敛率是次优的，由某个非线性方程的解决定。这使我们得以得出结论：在最坏情况下，任何程度的收缩操作都能改进匹配追踪的性能。