Efficient randomized tensor-based algorithms for function approximation and low-rank kernel interactions

TL;DR

This paper presents new randomized tensor algorithms for efficient multivariate function approximation and low-rank kernel matrix approximations, improving computational efficiency and accuracy in high-dimensional settings.

Contribution

It introduces novel randomized tensor compression techniques and applies them to develop efficient low-rank kernel approximations with detailed analysis and numerical validation.

Findings

Tensor compression reduces computational costs significantly.

Low-rank kernel approximations are efficient and scalable.

Numerical experiments confirm accuracy and efficiency.

Abstract

In this paper, we introduce a method for multivariate function approximation using function evaluations, Chebyshev polynomials, and tensor-based compression techniques via the Tucker format. We develop novel randomized techniques to accomplish the tensor compression, provide a detailed analysis of the computational costs, provide insight into the error of the resulting approximations, and discuss the benefits of the proposed approaches. We also apply the tensor-based function approximation to develop low-rank matrix approximations to kernel matrices that describe pairwise interactions between two sets of points; the resulting low-rank approximations are efficient to compute and store (the complexity is linear in the number of points). We have detailed numerical experiments on example problems involving multivariate function approximation, low-rank matrix approximations of kernel matrices involving well-separated clusters of sources and target points, and a global low-rank approximation of kernel matrices with an application to Gaussian processes.