Online Change-Point Detection with Persistent Laplacian Features

Abstract

Online change-point detection in high-dimensional nonlinear time series faces two challenges. The underlying distributions are difficult to model, and state changes are difficult to characterize. We propose persistent Laplacian cumulative sum (PL-CUSUM) to address these challenges. PL-CUSUM maps delay-embedded sliding windows to point clouds. It extracts persistent Betti vectors and the positive spectra of persistent Laplacians from the same Vietoris-Rips filtration. A ridge-whitened projection converts these features into a scalar score. Page's CUSUM recursion then accumulates this score over time. The positive spectra capture within-scale connectivity and geometric information that persistent Betti vectors do not record. Under a finite-support local model, we prove that the oracle upper bound on detection delay and a local minimax lower bound have the same order. This common order is the logarithm of the average run length constraint divided by the squared ridge-whitened separation. We also establish finite-horizon false-alarm and expected-delay bounds for plug-in whitened scores under weak dependence within each state. The method has two phases. Phase I estimates the projection parameters, selects the feature configuration, and calibrates the control limit. Phase II updates the resulting CUSUM statistic online. Experiments on simulated and real monitoring data show stable false-alarm control and competitive detection performance.

51 pages, 9 figures