基于降基方法的贪婪采样设计：一致范数下的最优恢复

中文摘要 AI 辅助

我们研究再生核希尔伯特空间（RKHS）中一致范数下的最优采样恢复问题。对每个具有有界核的RKHS，我们建立了线性采样宽度与Gelfand宽度之间的新比较，该比较克服了已知的平方根间隙，且无需测度或Christoffel型条件。我们的界依赖于通过（弱）P-贪婪点处的核插值得到的嵌套采样设计。在额外的（多项式）衰减假设下，Gelfand宽度的衰减率可直接传递到采样宽度。无论是对数过采样还是取Gelfand宽度的平方根，我们都能得到两者间的直接比较（无需衰减假设），这对超多项式衰减（如Paley-Wiener空间中的情况）尤为有效。我们的结果源于两种宽度在核平移意义下的表示，且在相反方向上得到了尖锐降基选择的新存在性结果。对Legendre、混合Sobolev及Paley-Wiener核的数值实验验证了我们的发现。

英文摘要

We study optimal sampling recovery in reproducing kernel Hilbert spaces (RKHS) in the uniform norm. For every RKHS with bounded kernel, we establish new comparisons between linear sampling widths and Gelfand widths that overcome the known square-root gap, without requiring a measure or a Christoffel-type condition. Our bounds rely on nested sampling designs obtained by kernel interpolation at (weak) P-greedy points. Under additional (polynomial) decay assumptions the decay rate of the Gelfand widths directly transfers to the sampling widths. With either a logarithmic oversampling or passing to the square root of the Gelfand widths we obtain a direct comparison (requiring no decay assumption) between them. This is particularly effective for super-polynomial decay, such as in Paley-Wiener spaces. Our results follow from representations of both widths in terms of kernel translates and yield, in the opposite direction, a new existence result for a sharp reduced basis selection. Numerical experiments for Legendre, mixed-Sobolev, and Paley-Wiener kernels illustrate our findings.

发表机构

• Chemnitz University of Technology（开姆尼茨工业大学）
• Institute of Mathematics of NAS of Ukraine（乌克兰国家科学院数学研究所）

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