论文标题
收敛速率的稳定性：非lipschitz域上的内插
Stability of convergence rates: Kernel interpolation on non-Lipschitz domains
论文作者
论文摘要
繁殖内核希尔伯特空间（RKHS）中核插值的错误估计通常在域的形状上具有相当限制的特性，尤其是在无限光滑的核（如流行的高斯内核）的情况下。 在本文中，我们利用对贪婪的内核算法的分析，以证明可以为任意域$ subset \ subset \ mathbb {r}^d $获得核插值的收敛结果（在插值点数）中，从而允许非lipschitz domains tomains。尖和不规则边界。尤其是我们表明，当进入较小的域$ \tildeΩ\ subsetω\ subset \ mathbb {r}^d $时，收敛速率不会恶化 - 即，收敛速率与进入子集相对于稳定。 该结果的影响在有限的核以及像高斯内核等无限平滑度的示例中解释了。绘制了Sobolev空间中近似值的比较，其中域$ω$的形状对近似属性有影响。数值实验说明并确认实验。
Error estimates for kernel interpolation in Reproducing Kernel Hilbert Spaces (RKHS) usually assume quite restrictive properties on the shape of the domain, especially in the case of infinitely smooth kernels like the popular Gaussian kernel. In this paper we leverage an analysis of greedy kernel algorithms to prove that it is possible to obtain convergence results (in the number of interpolation points) for kernel interpolation for arbitrary domains $Ω\subset \mathbb{R}^d$, thus allowing for non-Lipschitz domains including e.g. cusps and irregular boundaries. Especially we show that, when going to a smaller domain $\tildeΩ \subset Ω\subset \mathbb{R}^d$, the convergence rate does not deteriorate - i.e. the convergence rates are stable with respect to going to a subset. The impact of this result is explained on the examples of kernels of finite as well as infinite smoothness like the Gaussian kernel. A comparison to approximation in Sobolev spaces is drawn, where the shape of the domain $Ω$ has an impact on the approximation properties. Numerical experiments illustrate and confirm the experiments.
