Covariance scanning for adaptively optimal change point detection in high-dimensional linear models

This paper investigates the detection and estimation of a single change in high-dimensional linear models. We derive minimax lower bounds for the detection boundary and the estimation rate, which uncover a phase transition governed by the sparsity of the covariance-weighted differential parameter. This form of "inherent sparsity" captures a delicate interplay between the covariance structure of the regressors and the change in regression coefficients on the detectability of a change point. Complementing the lower bounds, we introduce two covariance scanning-based methods, McScan and QcSan, which achieve minimax optimal performance (up to possible logarithmic factors) in the sparse and the dense regimes, respectively. In particular, QcScan is the first method shown to achieve consistency in the dense regime and further, we devise a combined procedure which is adaptively minimax optimal across sparse and dense regimes without the knowledge of the sparsity. Computationally, covariance scanning-based methods avoid costly computation of Lasso-type estimators and attain worst-case computation complexity that is linear in the dimension and sample size. Additionally, we consider the post-detection estimation of the differential parameter and the refinement of the change point estimator. Simulation studies support the theoretical findings and demonstrate the computational and statistical efficiency of the proposed covariance scanning methods.