2021/02/12 by Christian Fröhlich, Alexandra Gessner, Fröhlich, Christian +7
Computer Science · Mathematics · Physics and Astronomy · #Algorithm #Applied mathematics #Artificial intelligence #Bayesian probability #Computer science #FOS: Computer and information sciences #Gaussian Processes and Bayesian Inference #Generative Adversarial Networks and Image Synthesis #Geometry #Information geometry #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Mathematical analysis #Mathematical optimization #Mathematics #Metric (unit) #Model Reduction and Neural Networks #Nonlinear system #Numerical integration #Physics #Probabilistic logic #Quadrature (astronomy) #Range (aeronautics) #Riemannian geometry #Scalar curvature #cs.LG #stat.ML
paper · pdf · doi:10.48550/arxiv.2102.06645
published in arXiv (Cornell University), 3459-3468 (Cornell University)
openalex publication_date 2021/02/12 · arxiv created 2021/06/10 · arxiv updated 2021/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
Riemannian manifolds provide a principled way to model nonlinear geometric structure inherent in data. A Riemannian metric on said manifolds determines geometry-aware shortest paths and provides the means to define statistical models accordingly. However, these operations are typically computationally demanding. To ease this computational burden, we advocate probabilistic numerical methods for Riemannian statistics. In particular, we focus on Bayesian quadrature (BQ) to numerically compute integrals over normal laws on Riemannian manifolds learned from data. In this task, each function evaluation relies on the solution of an expensive initial value problem. We show that by leveraging both prior knowledge and an active exploration scheme, BQ significantly reduces the number of required evaluations and thus outperforms Monte Carlo methods on a wide range of integration problems. As a concrete application, we highlight the merits of adopting Riemannian geometry with our proposed framework on a nonlinear dataset from molecular dynamics.