High-Dimensional Robust Change-Point Detection via Angular Kernel Statistics

Abstract

We study nonparametric change-point detection for high-dimensional data in regimes where inference must be performed from small batches of observations. Our primary focus is the high-dimensional, low sample size (HDLSS) regime, where the sequence length is fixed while the ambient dimension diverges. We propose a dimension-averaged angular kernel scan framework for detecting marginal distributional shifts. The statistic aggregates bounded one-dimensional angular discrepancies across coordinates, yielding a fully nonparametric, hyperparameter-free, and moment-agnostic estimator that remains well-defined without specifying, estimating, or assuming finite marginal moments; for example, under heavy-tailed or contaminated distributions. For the offline single-change problem, we derive an exact population mean factorization into a universal deterministic shape function and a scalar signal factor, and characterize the exact null covariance structure up to a scalar variance factor, both valid for any fixed sample size and dimension. We also establish an HDLSS multivariate central limit theorem under cross-coordinate strong mixing which leads to a variance-calibrated asymptotically distribution-free test, asymptotic type-I error control, and lower bounds on power and localization accuracy. We further extend the offline procedure to a fixed-window sequential monitoring procedure for high-dimensional streaming data, and obtain ARL calibration and worst-case Pollak EDD bounds. Simulation studies demonstrate that the proposed method can accurately detect and localize changes in many challenging HDLSS and streaming high-dimensional settings where moment-based or hyperparameter-sensitive procedures may be extremely unstable or inaccurate.