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.

翻译：我们研究了匹配追踪（即纯贪婪算法）在通过字典中元素的稀疏线性组合逼近目标函数时的基本极限。当目标函数属于字典对应的变差空间时，过去数十年的许多重要工作已获得匹配追踪收敛速率的上下界，但这些界并不匹配。本文的主要贡献是填补这一空白，刻画出匹配追踪性能的尖锐特征。我们通过改进现有下界使其与最优上界匹配来实现这一目标。具体而言，我们构造了一个最坏情况字典，证明现有上界无法进一步改进。结果发现，与其他贪婪算法变体不同，收敛速率是次优的，并取决于某个非线性方程的解。这使得我们能够得出结论：在最坏情况下，任何程度的收缩都能改进匹配追踪。