Multiresolution kernel matrix algebra

TL;DR

This paper introduces a sparse algebra for samplet compressed kernel matrices, enabling efficient analysis of scattered data with linear scaling in cost and memory, and supports matrix operations and functions.

Contribution

It develops a novel S-format for kernel matrices that is optimally sparse, supports efficient operations, and extends to matrix functions, with applications in Gaussian process learning.

Findings

Achieves linear scaling in cost and memory for kernel matrix compression.

Supports efficient addition, multiplication, and matrix functions in S-format.

Demonstrates effectiveness in Gaussian process learning for spatial statistics.

Abstract

We propose a sparse algebra for samplet compressed kernel matrices, to enable efficient scattered data analysis. We show the compression of kernel matrices by means of samplets produces optimally sparse matrices in a certain S-format. It can be performed in cost and memory that scale essentially linearly with the matrix size $N$, for kernels of finite differentiability, along with addition and multiplication of S-formatted matrices. We prove and exploit the fact that the inverse of a kernel matrix (if it exists) is compressible in the S-format as well. Selected inversion allows to directly compute the entries in the corresponding sparsity pattern. The S-formatted matrix operations enable the efficient, approximate computation of more complicated matrix functions such as ${\bm A}^\alpha$ or $\exp({\bm A})$. The matrix algebra is justified mathematically by pseudo differential calculus. As an application, efficient Gaussian process learning algorithms for spatial statistics is considered. Numerical results are presented to illustrate and quantify our findings.