Online Kernel CUSUM for Change-Point Detection

We develop an online kernel Cumulative Sum (CUSUM) procedure, which consists of a parallel set of kernel statistics with different window sizes to account for the unknown change-point location. Compared with many existing sliding window-based kernel change-point detection procedures, which correspond to the Shewhart chart-type procedure, the proposed procedure is more sensitive to small changes. We further present a recursive computation of detection statistics, which is crucial for online procedures to achieve a constant computational and memory complexity, such that we do not need to calculate and remember the entire Gram matrix, which can be a computational bottleneck otherwise. We obtain precise analytic approximations of the two fundamental performance metrics, the Average Run Length (ARL) and Expected Detection Delay (EDD). Furthermore, we establish the optimal window size on the order of $\log ({\rm ARL})$ such that there is nearly no power loss compared with an oracle procedure, which is analogous to the classic result for window-limited Generalized Likelihood Ratio (GLR) procedure. We present extensive numerical experiments to validate our theoretical results and the competitive performance of the proposed method.