Manifold Learning is a set of machine learning methods that model high-dimensional data as lying near a lower-dimensional manifold embedded in a larger space. It is used for nonlinear dimensionality reduction, visualization, and feature extraction when linear methods are insufficient. Common approaches include Isomap, LLE, and diffusion maps, and are applied in computer vision, speech, and scientific data analysis.

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Correlation-aware Online Change Point Detection

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Change point detection aims to identify abrupt shifts occurring at multiple points within a data sequence. This task becomes particularly challenging in the online setting, where different types of change can occur, including shifts in both the marginal and joint distributions of the data. In this paper, we address these challenges by tracking the Riemannian geometry of correlation matrices, allowing Riemannian metrics to compute the geodesic distance as an accurate measure of correlation dynamics.We introduce Rio-CPD, a correlation-aware online change point detection framework that integrates the Riemannian geometry of the manifold of symmetric positive definite matrices with the cumulative sum (CUSUM) statistic for detecting change points. Rio-CPD employs a novel CUSUM design by computing the geodesic distance between current observations and the Fréchet mean of prior observations. With appropriate choices of Riemannian metrics, Rio-CPD offers a simple yet effective and computationally efficient algorithm. We also provide a theoretical analysis on standard metrics for change point detection within Rio-CPD. Experimental results on both synthetic and real-world datasets demonstrate that Rio-CPD outperforms existing methods on detection accuracy, average detection delay, and efficiency.

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