We present new convergence estimates of generalized empirical interpolation methods in terms of the entropy numbers of the parametrized function class. Our analysis is transparent and leads to sharper convergence rates than the classical analysis via the Kolmogorov n-width. In addition, we also derive novel entropy-based convergence estimates of the Chebyshev greedy algorithm for sparse n-term nonlinear approximation of a target function. This also improves classical convergence analysis when corresponding entropy numbers decay fast enough.

翻译：我们基于参数化函数类的熵数提出了广义经验插值方法的新收敛估计。该分析方法清晰透明，相较于通过科尔莫戈罗夫n-宽度的经典分析，能推导出更优的收敛速率。此外，我们还针对目标函数的稀疏n项非线性逼近，推导出切比雪夫贪心算法基于熵数的新收敛估计。当对应熵数以足够快的速度衰减时，这一结果同样改进了经典收敛分析。