Minimax rates in variance and covariance changepoint testing
Per August Jarval Moen
TL;DR
The paper delivers a precise minimax characterization for changepoints in variance and covariance under independent sub-Gaussian data. It establishes a univariate rate $\rho^*(n) \asymp \log\log(8n)$ and a sparsity-aware multivariate lower bound driven by $s$ and $p$, then provides matching (where feasible) upper bounds via sparse-eigenvalue tests and adaptive procedures. To address computation in high dimensions, it develops a convex SDP-based relaxation that yields a scalable, adaptive changepoint test. The results reveal when changepoint detection is statistically feasible and quantify the tradeoffs between signal strength, sparsity, sample size, and computation, contributing both theory and practical methodology for covariance changepoint analysis.
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.