Data sparse multilevel covariance estimation in optimal complexity

TL;DR

This paper introduces a data sparse multilevel covariance estimator using $ ext{H}^2$-approximation for efficient covariance estimation in high-dimensional problems, achieving optimal complexity and handling large datasets.

Contribution

It generalizes asymptotically smooth kernels to Gevrey classes, develops variable-order $ ext{H}^2$-approximations, and designs a linear complexity multilevel estimator for large covariance matrices.

Findings

Achieves quadratic complexity for multilevel covariance estimation.

Handles covariance matrices with tens of billions of entries.

Demonstrates efficiency through numerical examples.

Abstract

We consider the $\mathcal{H}^2$-formatted compression and computational estimation of covariance functions on a compact set in $\mathbb{R}^d$. The classical sample covariance or Monte Carlo estimator is prohibitively expensive for many practically relevant problems, where often approximation spaces with many degrees of freedom and many samples for the estimator are needed. In this article, we propose and analyze a data sparse multilevel sample covariance estimator, i.e., a multilevel Monte Carlo estimator. For this purpose, we generalize the notion of asymptotically smooth kernel functions to a Gevrey type class of kernels for which we derive new variable-order $\mathcal{H}^2$-approximation rates. These variable-order $\mathcal{H}^2$-approximations can be considered as a variant of $hp$-approximations. Our multilevel sample covariance estimator then uses an approximate multilevel hierarchy of variable-order $\mathcal{H}^2$-approximations to compress the sample covariances on each level. The non-nestedness of the different levels makes the reduction to the final estimator nontrivial and we present a suitable algorithm which can handle this task in linear complexity. This allows for a data sparse multilevel estimator of Gevrey covariance kernel functions in the best possible complexity for Monte Carlo type multilevel estimators, which is quadratic. Numerical examples which estimate covariance matrices with tens of billions of entries are presented.