Related papers: Scaling of Radial Basis Functions
We examine the necessity of interpolation in overparameterized models, that is, when achieving optimal predictive risk in machine learning problems requires (nearly) interpolating the training data. In particular, we consider simple…
Multiresolution analyses based upon interpolets, interpolating scaling functions introduced by Deslauriers and Dubuc, are particularly well-suited to physical applications because they allow exact recovery of the multiresolution…
I develop a weight func theory of zero order basis func interpolants and smoothers.**Ch1 Basis funcs and data spaces are defined using wt funcs. Data (native)spaces are used to formulate the variational problems which define our…
Multiphysics simulations frequently require transferring solution fields between subproblems with non-matching spatial discretizations, typically using interpolation techniques. Standard methods are usually based on measuring the closeness…
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 ridge regression is well-known to achieve minimax optimal rates in low-dimensional settings. However, its behavior in high dimensions is much less understood. Recent work establishes consistency for kernel regression under certain…
We present the first optimal rates for infinite-dimensional vector-valued ridge regression on a continuous scale of norms that interpolate between $L_2$ and the hypothesis space, which we consider as a vector-valued reproducing kernel…
In this paper, we study functional approximations where we choose the so-called radial basis function method and more specifically, quasi-interpolation. From the various available approaches to the latter, we form new quasi-Lagrange…
We propose and study a new quasi-interpolation method on spheres featuring the following two-phase construction and analysis. In Phase I, we analyze and characterize a large family of zonal kernels (e.g., the spherical version of Poisson…
We study the interpolation power of deep ReLU neural networks. Specifically, we consider the question of how efficiently, in terms of the number of parameters, deep ReLU networks can interpolate values at $N$ datapoints in the unit ball…
A kernel based procedure for correcting experimental data for distortions due to the finite resolution and limited detector acceptance is presented. The unfolding problem is known to be an ill-posed problem that can not be solved without…
Approximation/interpolation from spaces of positive definite or conditionally positive definite kernels is an increasingly popular tool for the analysis and synthesis of scattered data, and is central to many meshless methods. For a set of…
Generalization beyond a training dataset is a main goal of machine learning, but theoretical understanding of generalization remains an open problem for many models. The need for a new theory is exacerbated by recent observations in deep…
We investigate the problem of approximating the matrix function $f(A)$ by $r(A)$, with $f$ a Markov function, $r$ a rational interpolant of $f$, and $A$ a symmetric Toeplitz matrix. In a first step, we obtain a new upper bound for the…
We study the problem of the appropriate choice of the interpolating kernel to be used in the evaluation of gradients of functions. Such interpolation technique is often used in applications, e.g. it is typical for Smoothed Particle…
The kernel-based multi-scale method has been proven to be a powerful approximation method for scattered data approximation problems which is computationally superior to conventional kernel-based interpolation techniques. The multi-scale…
As modern machine learning models continue to advance the computational frontier, it has become increasingly important to develop precise estimates for expected performance improvements under different model and data scaling regimes.…
Structured Kernel Interpolation (SKI) (Wilson et al. 2015) helps scale Gaussian Processes (GPs) by approximating the kernel matrix via interpolation at inducing points, achieving linear computational complexity. However, it lacks rigorous…
We analyze the approximation by radial basis functions of a hypersingular integral equation on an open surface. In order to accommodate the homogeneous essential boundary condition along the surface boundary, scaled radial basis functions…
The usual power function error estimates do not capture the true order of uniform accuracy for thin plate spline interpolation to smooth data functions in one variable. In this paper we propose a new type of power function and we show,…