We present convergence estimates of two types of greedy algorithms in terms of the metric entropy of underlying compact sets. In the first part, we measure the error of a standard greedy reduced basis method for parametric PDEs by the metric entropy of the solution manifold in Banach spaces. This contrasts with the classical analysis based on the Kolmogorov n-widths and enables us to obtain direct comparisons between the greedy algorithm error and the entropy numbers, where the multiplicative constants are explicit and simple. The entropy-based convergence estimate is sharp and improves upon the classical width-based analysis of reduced basis methods for elliptic model problems. In the second part, we derive a novel and simple convergence analysis of the classical orthogonal greedy algorithm for nonlinear dictionary approximation using the metric entropy of the symmetric convex hull of the dictionary. This also improves upon existing results by giving a direct comparison between the algorithm error and the metric entropy.

翻译：我们基于底层紧集的度量熵，给出了两类贪婪算法的收敛估计。在第一部分中，我们通过巴拿赫空间中解流形的度量熵来衡量参数化偏微分方程的标准贪婪降阶基方法的误差。这与基于柯尔莫哥洛夫n-宽度的经典分析形成对比，并使我们能够直接比较贪婪算法误差与熵数之间的关系，其中乘法常数明确且简单。该基于熵的收敛估计是精确的，并且改进了椭圆模型问题中降阶基方法的经典宽度分析。在第二部分中，我们利用字典的对称凸包的度量熵，导出了一种新颖且简单的经典正交贪婪算法用于非线性字典逼近的收敛分析。这还通过直接比较算法误差与度量熵，改进了现有结果。