Exploiting the variational interpretation of kernel interpolation we exhibit a direct connection between interpolation and regression, where interpolation appears as a limiting case of regression. By applying this framework to point clouds or samples of smooth manifolds (hypersurfaces, in particular), we show how fundamental geometric quantities such as tangent plane and principal curvatures can be computed numerically using a kernel based (approximate) level set function (often a defining function) for smooth hypersurfaces. In the case of point clouds, the approach generates an interpolated hypersurface, which is an approximation of the underlying manifold when the cloud is a sample of it. It is shown how the geometric quantities obtained can be used in the numerical approximation/computation of geometric operators like the surface gradient or the Laplace-Beltrami operator in the spirit of kernel based meshfree methods. Kernel based interpolation can be extremely ill-posed, especially when using smooth kernels, and the regression approximation offers a natural regularization that proves also quite useful when dealing with geometric or functional data that are affected by errors or noise.
In this article, we use the knowledge of positive definite tensors to develop a concept of positive definite multi-kernels to construct the kernel-based interpolants of scattered data. By the techniques of reproducing kernel Banach spaces, the optimal recoveries and error analysis of the kernel-based interpolants are shown for a special class of strictly positive definite multi-kernels.
We construct approximate Fekete point sets for kernel-based interpolation by maximising the determinant of a kernel Gram matrix obtained via truncation of an orthonormal expansion of the kernel. Uniform error estimates are proved for kernel interpolants at the resulting points. If the kernel is Gaussian we show that the approximate Fekete points in one dimension are the solution to a convex optimisation problem and that the interpolants converge with a super-exponential rate. Numerical examples are provided for the Gaussian kernel.
Lappeenranta–Lahti University of Technology LUT（拉彭兰塔-拉赫蒂理工大学）；Ca’ Foscari University of Venice（威尼斯卡福斯卡里大学）；Ludwig Maximilian University of Munich（慕尼黑路德维希-马克西米利安大学）
A mixed interpolation-regression method for function approximation on certain planar domains
La Laguna University, Spain（拉帕尔马大学）；IMAG and Departamento de Matemática Aplicada, Universidad de Granada, Spain（格拉纳达大学）；Departamento de Matemáticas, Universidad Carlos III de Madrid, Spain（马德里卡洛斯三世大学）
The paper is concerned with classic kernel interpolation methods, in addition to approximation methods that are augmented by gradient measurements. To apply kernel interpolation using radial basis functions (RBFs) in a stable way, we propose a type of quasi-optimal interpolation points, searching from a large set of \textit{candidate} points, using a procedure similar to designing Fekete points or power function maximizing points that use pivot from a Cholesky decomposition. The proposed quasi-optimal points result in a smaller condition number, and thus mitigates the instability of the interp…
Institut für Numerische Simulation, Universität Bonn（波恩大学数值模拟研究所）；Fraunhofer Institute for Algorithms and Scientific Computing (SCAI)（弗劳恩霍夫算法与科学计算研究所）；Departement Mathematik und Informatik, Universität Basel（巴塞尔大学数学与信息学系）；Istituto Dalle Molle di studi sull’intelligenza artificiale, Università della Svizzera italiana（意大利瑞士大学达莱莫勒人工智能研究学院）
A Unified Framework for Efficient Kernel and Polynomial Interpolation
Kahlert School of Computing, University of Utah（犹他大学卡勒特计算学院）；Department of Applied Mathematics and Statistics, Colorado School of Mines（科罗拉多矿业大学应用数学与统计系）；Department of Mathematics, and Scientific Computing and Imaging (SCI) Institute, University of Utah（犹他大学数学系及科学计算与成像研究所）；Stena Center for Financial Technology, University of Utah（犹他大学斯泰纳金融技术研究中心）
Fast Multipole Methods (FMMs) based on the oscillatory Helmholtz kernel can reduce the cost of solving N-body problems arising from Boundary Integral Equations (BIEs) in acoustic or electromagnetics. However, their cost strongly increases in the high-frequency regime. This paper introduces a new directional FMM for oscillatory kernels (defmm - directional equispaced interpolation-based fmm), whose precomputation and application are FFT-accelerated due to polynomial interpolations on equispaced grids. We demonstrate the consistency of our FFT approach, and show how symmetries can be exploited i…
Matrices resulting from the discretization of a kernel function, e.g., in the context of integral equations or sampling probability distributions, can frequently be approximated by interpolation. In order to improve the efficiency, a multi-level approach can be employed that involves interpolating the kernel function and its approximations multiple times. This article presents a new approach to analyze the error incurred by these iterated interpolation procedures that is considerably more elegant than its predecessors and allows us to treat not only the kernel function itself, but also its der…
Quasi-interpolation using generalized Gaussian kernels
School of Big Data and Statistics, Anhui University（安徽大学大数据与统计学院）；School of Mathematics and Statistics, Nanjing University of Science and Technology（南京理工大学数学与统计学院）
For $h>0$ and positive integers $m$, $d$, such that $m>d/2$, we study non-stationary interpolation at the points of the scaled grid $h\mathbb{Z}^d$ via the Matérn kernel $Φ_{m,d}$---the fundamental solution of $(1-Δ)^m$ in $\mathbb{R}^d$. We prove that the Lebesgue constants of the corresponding interpolation operators are uniformly bounded as $h\to0$ and deduce the convergence rate $O(h^{2m})$ for the scaled interpolation scheme. We also provide convergence results for approximation with Matérn and related compactly supported polyharmonic kernels.
In this paper we propose an enhanced version of the residual sub-sampling method (RSM) in [9] for adaptive interpolation by radial basis functions (RBFs). More precisely, we introduce in the context of sub-sampling methods a maximum profile likelihood estimation (MPLE) criterion for the optimal selection of the RBF shape parameter. This choice is completely automatic, provides highly reliable and accurate results for any RBFs, and, unlike the original RSM, guarantees that the RBF interpolant exists uniquely. The efficacy of this new method, called MPLE-RSM, is tested by numerical experiments o…
Data sites selected from modeling high-dimensional problems often appear scattered in non-paternalistic ways. Except for sporadic clustering at some spots, they become relatively far apart as the dimension of the ambient space grows. These features defy any theoretical treatment that requires local or global quasi-uniformity of distribution of data sites. Incorporating a recently-developed application of integral operator theory in machine learning, we propose and study in the current article a new framework to analyze kernel interpolation of high dimensional data, which features bounding stoc…
This paper presents a general high-order kernel regularization technique applicable to all four integral operators of Calderón calculus associated with linear elliptic PDEs in two and three spatial dimensions. Like previous density interpolation methods, the proposed technique relies on interpolating the density function around the kernel singularity in terms of solutions of the underlying homogeneous PDE, so as to recast singular and nearly singular integrals in terms of bounded (or more regular) integrands. We present here a simple interpolation strategy which, unlike previous approaches, do…
Refined rates of convergence for target-data dependent greedy generalized interpolation with Sobolev kernels
Institute of Applied Analysis and Numerical Simulation, University of Stuttgart（斯图加特大学应用分析与数值模拟研究所）；Department of Environmental Sciences, Informatics and Statistics, Ca’ Foscari University of Venice（威尼斯卡福斯卡里大学环境科学、信息学与统计系）；Department of Mathematics, Ludwig-Maximilians-Universität München（慕尼黑路德维希-马克西米利安大学数学系）；Institute for Stochastics and Applications, University of Stuttgart（斯图加特大学随机过程与应用研究所）
A mixed interpolation-regression method for numerical integration on the unit circle using zeros of para-orthogonal polynomials
La Laguna University, Spain（拉戈哈大学）；IMAG and Departamento de Matemática Aplicada, Universidad de Granada, Spain（格拉纳达大学）；Departamento de Matemáticas, Universidad Carlos III de Madrid, Spain（马德里卡洛斯三世大学）
This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice -- a setting that, as pointed out by Zeng, Leung, Hickernell (MCQMC2004, 2006) and Zeng, Kritzer, Hickernell (Constr. Approx., 2009), allows fast evaluation by fast Fourier transform, so avoiding the need for a linear solver. The main contribution of the paper is the application to the approximation problem for uncertainty quantification of elliptic partial differential equations, with the diffusion coefficient given by a random field that is periodi…
Università degli Studi di Padova（帕多瓦大学）；Politecnico di Torino（都灵理工大学）；FAR Networks S.r.l.（FAR Networks有限责任公司）；Fondazione Bruno Kessler（布鲁诺·凯斯勒基金会）
The polynomial kernels are widely used in machine learning and they are one of the default choices to develop kernel-based classification and regression models. However, they are rarely used and considered in numerical analysis due to their lack of strict positive definiteness. In particular they do not enjoy the usual property of unisolvency for arbitrary point sets, which is one of the key properties used to build kernel-based interpolation methods. This paper is devoted to establish some initial results for the study of these kernels, and their related interpolation algorithms, in the conte…
