Multiresolution kernel matrix algebra

• URL: http://arxiv.org/abs/2211.11681v2
• Date: Wed, 3 May 2023 21:27:15 GMT
• Title: Multiresolution kernel matrix algebra
• Authors: H. Harbrecht, M. Multerer, O. Schenk, and Ch. Schwab
• Abstract summary: We show the compression of kernel matrices by means of samplets produces optimally sparse matrices in a certain S-format. The inverse of a kernel matrix (if it exists) is compressible in the S-format as well. The matrix algebra is justified mathematically by pseudo differential calculus.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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