SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2249-2275, October 2024.
Abstract. Kernel interpolation is a versatile tool for the approximation of functions from data, and it can be proven to have some optimality properties when used with kernels related to certain Sobolev spaces. In the context of interpolation, the selection of optimal function sampling locations is a central problem, both from a practical perspective and as an interesting theoretical question. Greedy interpolation algorithms provide a viable solution for this task, being efficient to run and provably accurate in their approximation. In this paper we close a gap that is present in the convergence theory for these algorithms by employing a recent result on general greedy algorithms. This modification leads to new convergence rates which match the optimal ones when restricted to the [math]-greedy target-data-independent selection rule and can additionally be proven to be optimal when they fully exploit adaptivity ([math]-greedy). Other than closing this gap, the new results have some significance in the broader setting of the optimality of general approximation algorithms in reproducing kernel Hilbert spaces, as they allow us to compare adaptive interpolation with nonadaptive best nonlinear approximation.

中文翻译：

---

Sobolev再生核希尔伯特空间中目标数据相关核贪婪插值的最优性


《SIAM 数值分析杂志》，第 62 卷，第 5 期，第 2249-2275 页，2024 年 10 月。

抽象的。核插值是一种用于根据数据逼近函数的多功能工具，并且当与与某些 Sobolev 空间相关的核一起使用时，可以证明它具有一些最优性属性。在插值的背景下，无论从实践角度还是作为一个有趣的理论问题，最优函数采样位置的选择都是一个中心问题。贪心插值算法为该任务提供了一个可行的解决方案，运行效率高并且近似值可证明是准确的。在本文中，我们通过采用一般贪婪算法的最新结果来弥补这些算法的收敛理论中存在的差距。这种修改导致了新的收敛速度，当限制于[数学]-贪婪的目标数据独立选择规则时，该收敛速度与最佳收敛速度相匹配，并且当它们充分利用自适应性（[数学]-贪婪）时，还可以证明是最佳的。除了缩小这一差距之外，新结果对于再生核希尔伯特空间中通用逼近算法的最优性的更广泛设置具有一定意义，因为它们允许我们将自适应插值与非自适应最佳非线性逼近进行比较。