Riemannian Change Point Detection on Manifolds with Robust Centroid Estimation

TL;DR

This paper introduces a robust Riemannian change point detection method using M-estimation for centroid estimation, improving detection robustness on manifold-valued streaming data.

Contribution

It proposes a novel robust centroid estimation approach on manifolds using M-estimation, and a stochastic optimization algorithm for efficient computation.

Findings

Robust centroid estimation improves change point detection accuracy.

The method outperforms traditional approaches on simulated and real data.

It demonstrates effectiveness across different Riemannian manifolds.

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.