Asymptotic analysis in multivariate average case approximation with Gaussian kernels

We consider tensor product random fields Y_d, d∈ℕ, whose covariance funtions are Gaussian kernels. The average case approximation complexity n^Y_d(ε) is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate Y_d, with relative 2-average error not exceeding a given threshold ε∈(0,1). We investigate the growth of n^Y_d(ε) for arbitrary fixed ε∈(0,1) and d→∞. Namely, we find criteria of boundedness for n^Y_d(ε) on d and of tending n^Y_d(ε)→∞, d→∞, for any fixed ε∈(0,1). In the latter case we obtain necessary and sufficient conditions for the following logarithmic asymptotics ln n^Y_d(ε)= a_d+q(ε)b_d+o(b_d), d→∞, with any ε∈(0,1). Here q (0,1)→ℝ is a non-decreasing function, (a_d)_d∈ℕ is a sequence and (b_d)_d∈ℕ is a positive sequence such that b_d→∞, d→∞. We show that only special quantiles of self-decomposable distribution functions appear as functions q in a given asymptotics.

There are no comments yet.

Authors

• 1 publication
• 1 publication
07/01/2019

Average case tractability of additive random fields with Korobov kernels

We investigate average case tractability of approximation of additive ra...
05/18/2020

Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces

Optimal linear prediction (also known as kriging) of a random field {Z(x...
10/12/2021

Countable Tensor Products of Hermite Spaces and Spaces of Gaussian Kernels

In recent years finite tensor products of reproducing kernel Hilbert spa...
01/31/2020

Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals

We study the approximation of compact linear operators defined over cert...
01/23/2021

The Gauss Hypergeometric Covariance Kernel for Modeling Second-Order Stationary Random Fields in Euclidean Spaces: its Compact Support, Properties and Spectral Representation

This paper presents a parametric family of compactly-supported positive ...
12/15/2020

Approximation by linear combinations of translates of a single function

We study approximation by arbitrary linear combinations of n translates ...
10/04/2018

Gaussian approximation of Gaussian scale mixture

For a given positive random variable V>0 and a given Z∼ N(0,1) independe...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.