Statistically and Computationally Efficient Change Point Localization in Regression Settings.

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. Specically, we assume we observe fxt; ytgn t=1 where fxtgn t=1 are p-dimensional covariates, fytgn t=1 are the univariate responses satisfying E(yt) = x> t for 1 tn and ft gn t=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 signicant improvement from the best rate Op(1= p 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 ft gn t=1 is small. [ABSTRACT FROM AUTHOR]