We study the detection of a change in the spatial 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.

翻译：我们研究了$n$个独立$p$维次高斯随机变量空间协方差矩阵的变化检测问题。第一项贡献是证明当$p=1$时，$\log\log(8n)$是方差变化的精确极小极大检验速率，从而完整刻画了单变量数据问题的特征。第二项贡献是在算子范数下推导了极小极大检验速率的下界，并考虑了特定的稀疏性概念。在参数空间的低维至中维区域，我们能够通过基于稀疏特征值的最优检验匹配上述下界。在剩余的高维参数空间区域，极小极大下界表明变化点检测极为困难。作为第三项贡献，我们提出了协方差变化最优多变量检验的计算可行变体，该检验同时自适应于名义噪声水平与变化的稀疏程度。