Statistically and Computationally Efficient Change Point Localization in Regression Settings

Wang, Daren; Zhao, Zifeng; Lin, Kevin; Willett, Rebecca

Detecting when the underlying distribution changes for the observed time series is a fundamental problem arising in a broad spectrum of applications. In this paper, we study multiple change-point localization in the high-dimensional regression setting, which is particularly challenging as no direct observations of the parameter of interest is available. Specifically, we assume we observe {xt,yt}nt=1 where {xt}nt=1 are p-dimensional covariates, {yt}nt=1 are the univariate responses satisfying 𝔼(yt)=x⊤tβ∗t for 1≤t≤n and {β∗t}nt=1 are the unobserved regression coefficients that change over time in a piecewise constant manner. We propose a novel projection-based algorithm, Variance Projected Wild Binary Segmentation~(VPWBS), which transforms the original (difficult) problem of change-point detection in p-dimensional regression to a simpler problem of change-point detection in mean of a one-dimensional time series. VPWBS is shown to achieve sharp localization rate Op(1/n) up to a log factor, a significant improvement from the best rate Op(1/n‾√) known in the existing literature for multiple change-point localization in high-dimensional regression. Extensive numerical experiments are conducted to demonstrate the robust and favorable performance of VPWBS over two state-of-the-art algorithms, especially when the size of change in the regression coefficients {β∗t}nt=1 is small.

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