Haeran Cho (University of Bristol) – Adaptively optimal change point detection in high-dimensional linear models

This paper studies the detection of a change in high-dimensional linear models. We derive the minimax lower bounds on the detection boundary and the rate of estimation which exhibit a phase transition with respect to the sparsity of the covariance-weighted differential parameter, revealing a delicate interaction between the covariance of regressors and the change in regression parameters. We complement these results by proposing methods that achieve minimax (near-)optimality in the sparse and the dense regimes, respectively. Referred to as McScan and QcScan, they scan the maximum and the quadratic aggregations of the local covariances at strategically selected locations for change point detection; in particular, QcScan is the first method shown to achieve consistency in the dense regime. Further, a combined method is proposed which is adaptively optimal even when the sparsity is unknown, and we complete the study of the change point problem by considering post-detection estimation of the differential parameter and the refinement of the change point estimators. Numerical experiments confirm the new findings as well as demonstrating the computational and statistical efficiency of the covariance-scanning based methods. This is joint work with Tobias Kley and Housen Li (University of Göttingen).