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

Arvind K. Saibaba; Rachel Minster; Misha E. Kilmer

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arXiv · 2107.13107

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

Arvind K. Saibaba·Rachel Minster·Misha E. Kilmer

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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.

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BibTeX RIS

Arvind K. Saibaba, Rachel Minster, Misha E. Kilmer. 2021-07-27. Efficient randomized tensor-based algorithms for function approximation and low-rank kernel interactions. https://arxiv.org/abs/2107.13107

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