Sparse inverse Cholesky factorization of dense kernel matrices by greedy conditional selection

TL;DR

This paper introduces a greedy selection algorithm for constructing sparse inverse Cholesky factors of dense kernel matrices, improving computational efficiency and accuracy in Gaussian process applications.

Contribution

It develops a novel greedy conditional selection method that leverages mutual information, reducing complexity and enhancing sparse Cholesky factorization for kernel matrices.

Findings

Reduces complexity from O(N k^4) to O(N k^2) using partial Cholesky maintenance.

Achieves efficient multiple target selection with complexity O(N k^2 + N m^2 + m^3).

Improves Gaussian process regression and preconditioning over k-nearest neighbors methods.

Abstract

Dense kernel matrices resulting from pairwise evaluations of a kernel function arise naturally in machine learning and statistics. Previous work in constructing sparse approximate inverse Cholesky factors of such matrices by minimizing Kullback-Leibler divergence recovers the Vecchia approximation for Gaussian processes. These methods rely only on the geometry of the evaluation points to construct the sparsity pattern. In this work, we instead construct the sparsity pattern by leveraging a greedy selection algorithm that maximizes mutual information with target points, conditional on all points previously selected. For selecting $k$ points out of $N$, the naive time complexity is $\mathcal{O}(N k^4)$, but by maintaining a partial Cholesky factor we reduce this to $\mathcal{O}(N k^2)$. Furthermore, for multiple ($m$) targets we achieve a time complexity of $\mathcal{O}(N k^2 + N m^2 + m^3)$, which is maintained in the setting of aggregated Cholesky factorization where a selected point need not condition every target. We apply the selection algorithm to image classification and recovery of sparse Cholesky factors. By minimizing Kullback-Leibler divergence, we apply the algorithm to Cholesky factorization, Gaussian process regression, and preconditioning with the conjugate gradient, improving over $k$-nearest neighbors selection.