Optimal nonparametric multivariate change point detection and localization

Abstract: We study the multivariate nonparametric change point detection problem, where the data are a sequence of independent $p$-dimensional random vectors whose distributions are piecewise-constant with Lipschitz densities changing at unknown times, called change points. We quantify the size of the distributional change at any change point with the supremum norm of the difference between the corresponding densities. We are concerned with the localization task of estimating the positions of the change points. In our analysis, we allow for the model parameters to vary with the total number of time points, including the minimal spacing between consecutive change points and the magnitude of the smallest distributional change. We provide information-theoretic lower bounds on both the localization rate and the minimal signal-to-noise ratio required to guarantee consistent localization. We formulate a novel algorithm based on kernel density estimation that nearly achieves the minimax lower bound, save possibly for logarithm factors. We have provided extensive numerical evidence to support our theoretical findings.