Calibrated Persistent-Laplacian CUSUM for Online Change-Point Detection

We propose the Persistent Laplacian Cumulative Sum (PL-CUSUM), an online change-point detection method for high-dimensional nonlinear time series. The method converts sliding windows into point clouds and uses persistent Laplacian spectra to construct the monitoring score for the Page cumulative sum (Page-CUSUM) recursion. Compared with detectors based only on persistent-homology summaries, PL-CUSUM further uses spectral information to capture within-scale connectivity and geometric structure beyond homology counts. Theoretically, we analyze two key performance criteria: false-alarm control and detection delay. We derive false-alarm-delay bounds for the oracle detector and show that the plug-in whitened score still controls false alarms over a finite monitoring horizon. Methodologically, we provide a Phase I/Phase II procedure that performs parameter selection and control-limit calibration before online recursion. Experiments on simulated systems and real monitoring data show that PL-CUSUM provides stable false-alarm control and competitive detection performance.