We study the fundamental limits of matching pursuit, or the pure greedy algorithm, for approximating a target function $ f $ by a linear combination $f_n$ of $n$ 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 $\|f-f_n\|$ 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, $n^{-\alpha}$, 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 which converge at the optimal rate $ n^{-1/2}$, the convergence rate $n^{-\alpha}$ is suboptimal. Here, $\alpha \approx 0.182$ is determined by the solution to a certain non-linear equation.

翻译：我们研究了匹配追踪（或称纯贪心算法）用于通过字典中$n$个元素的线性组合$f_n$逼近目标函数$f$的基本极限。当目标函数包含在字典对应的变分空间中时，过去几十年的许多杰出工作已经获得了匹配追踪误差$\|f-f_n\|$的上界和下界，但它们并不匹配。本文的主要贡献在于弥合这一差距，并获得了匹配追踪衰减率$n^{-\alpha}$的尖锐刻画。具体而言，我们构造了一个最坏情况字典，表明现有的最佳上界无法显著改进。事实证明，与其他以最优速率$n^{-1/2}$收敛的贪心算法变体不同，匹配追踪的收敛速率$n^{-\alpha}$是次优的。其中，$\alpha \approx 0.182$由某个非线性方程的解确定。