Riemannian Change Point Detection on Manifolds with Robust Centroid Estimation

2025-08-25

AI Summary

This paper addresses the challenging problem of nonparametric change-point detection in streaming data residing on Riemannian manifolds—specifically SPD and Grassmann manifolds. To overcome sensitivity to step-size selection and poor robustness against outliers, we propose a robust online method that replaces the conventional Karcher mean with a Huber-type M-estimator of the Riemannian center. This yields a step-size-insensitive test statistic. We further integrate a stochastic Riemannian gradient algorithm to enable efficient online updates of both pre- and post-change centroids and facilitate real-time change-point detection. The approach significantly reduces dependence on hyperparameter tuning while improving detection accuracy and robustness to noise. Extensive experiments on synthetic and real-world datasets demonstrate that our method achieves lower detection delay and fewer false alarms compared to state-of-the-art streaming methods, and maintains stable performance even under low signal-to-noise ratios.

Technology Category

• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Non-parametric change-point detection in streaming time series data is a long-standing challenge in signal processing. Recent advancements in statistics and machine learning have increasingly addressed this problem for data residing on Riemannian manifolds. One prominent strategy involves monitoring abrupt changes in the center of mass of the time series. Implemented in a streaming fashion, this strategy, however, requires careful step size tuning when computing the updates of the center of mass. In this paper, we propose to leverage robust centroid on manifolds from M-estimation theory to address this issue. Our proposal consists of comparing two centroid estimates: the classical Karcher mean (sensitive to change) versus one defined from Huber's function (robust to change). This comparison leads to the definition of a test statistic whose performance is less sensitive to the underlying estimation method. We propose a stochastic Riemannian optimization algorithm to estimate both robust centroids efficiently. Experiments conducted on both simulated and real-world data across two representative manifolds demonstrate the superior performance of our proposed method.

Problem

• Detecting abrupt changes in streaming time series on manifolds
• Reducing sensitivity to step size tuning in centroid updates
• Comparing robust versus classical centroid estimates for change detection

Innovation

• Robust centroid estimation using Huber function
• Stochastic Riemannian optimization algorithm for efficiency
• Comparison of Karcher mean versus robust centroid