Minimax rates in variance and covariance changepoint testing

TL;DR

This paper characterizes the minimax detection rates for covariance changes in multivariate data, providing optimal tests in low to moderate dimensions and a feasible adaptive method for high dimensions.

Contribution

It establishes the exact minimax rate for variance change detection in univariate data and develops a computationally feasible adaptive test for multivariate covariance changes.

Findings

Exact minimax rate for variance change when p=1 is log log n.

Optimal sparse eigenvalue-based test in low to moderate dimensions.

High-dimensional covariance change detection is fundamentally difficult.

Abstract

We study the detection of a change in the covariance matrix of $n$ independent sub-Gaussian random variables of dimension $p$. Our first contribution is to show that $\log\log(8n)$ is the exact minimax testing rate for a change in variance when $p=1$, thereby giving a complete characterization of the problem for univariate data. Our second contribution is to derive a lower bound on the minimax testing rate under the operator norm, taking a certain notion of sparsity into account. In the low- to moderate-dimensional region of the parameter space, we are able to match the lower bound from above with an optimal test based on sparse eigenvalues. In the remaining region of the parameter space, where the dimensionality is high, the minimax lower bound implies that changepoint testing is very difficult. As our third contribution, we propose a computationally feasible variant of the optimal multivariate test for a change in covariance, which is also adaptive to the nominal noise level and the sparsity level of the change.