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.

翻译：本文研究高维线性模型中单一变点的检测与估计问题。我们推导了检测边界与估计速率的极小极大下界，揭示了由协方差加权差分参数稀疏性支配的相变现象。这种"固有稀疏性"捕捉了协变量协方差结构对回归系数变化可检测性的复杂交互作用。作为下界的补充，我们提出了两种基于协方差扫描的方法——McScan与QcSan，它们分别在稀疏和密集状态下达到（至多对数因子差异的）极小极大最优性能。特别地，QcScan是首个被证明在密集状态下具有一致性的方法，进一步我们设计了一种组合程序，该程序在无需知晓稀疏性的情况下，在稀疏与密集状态间自适应达到极小极大最优。在计算方面，基于协方差扫描的方法避免了Lasso型估计量的昂贵计算，其最坏情况计算复杂度与维度和样本量呈线性关系。此外，我们研究了变点检测后差分参数的估计与变点估计量的精炼问题。模拟研究支持了理论发现，并验证了所提协方差扫描方法的计算效率和统计有效性。