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：本文研究了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和估计速率的最小最大下界，揭示了由协方差加权差分参数稀疏性控制的相变现象。这种"固有稀疏性"形式捕捉了回归变量协方差结构与回归系数变化之间对变点可检测性的微妙相互作用。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和QcScan，它们分别在稀疏和稠密机制下实现了最小最大最优性能（可能相差对数因子）。特别地，QcScan是首个被证明能在稠密机制下达到一致性的方法，并且我们进一步设计了无需稀疏性先验知识即可在稀疏和稠密机制间自适应达到最小最大最优的联合流程。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，获得了维度与样本量呈线性关系的最坏情况计算复杂度。此外，我们还考虑了差分参数的检测后估计以及变点估计量的精细化问题。仿真研究支持了理论发现，并证明了所提协方差扫描方法的计算与统计效率。