Geometry focuses on the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. It deals with the study of space and the shapes that inhabit it. Geometry plays a fundamental role in various fields, including physics, engineering, architecture, computer graphics, and many other scientific and technical disciplines.

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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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Fusing the Old with the New: Learning Relative Pose with Geometry-Guided Uncertainty

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Learning methods for relative camera pose estimation have been developed largely in isolation from classical geometric approaches. The question of how to integrate predictions from deep neural networks (DNNs) and solutions from geometric solvers, such as the 5-point algorithm [37], has as yet remained under-explored. In this paper, we present a novel framework that involves probabilistic fusion between the two families of predictions during network training, with a view to leveraging their complementary benefits in a learnable way. The fusion is achieved by learning the DNN un- certainty under explicit guidance by the geometric uncertainty, thereby learning to take into account the geometric solution in relation to the DNN prediction. Our network features a self-attention graph neural network, which drives the learning by enforcing strong interactions between different correspondences and potentially modeling complex relationships between points. We propose motion parmeterizations suitable for learning and show that our method achieves state-of-the-art performance on the challenging DeMoN [61] and ScanNet [8] datasets. While we focus on relative pose, we envision that our pipeline is broadly applicable for fusing classical geometry and deep learning.

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