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. Specifically, we assume we observe $\{ x_t, y_t\}_{t=1}^n$ where $\{ x_t\}_{t=1}^n$ are $p$-dimensional covariates, $\{y_t\}_{t=1}^n$ are the univariate responses satisfying $\mathbb{E}(y_t) = x_t^\top \beta_t^* \text{ for } 1\le t \le n$ and $\{\beta_t^*\}_{t=1}^n$ 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 $O_p(1/n)$ up to a log factor, a significant improvement from the best rate $O_p(1/\sqrt{n})$ 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 $\{\beta_t^*\}_{t=1}^n$ is small.