Bayesian Quadrature on Riemannian Data Manifolds

02/12/2021
by   Christian Fröhlich, et al.
14

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.

READ FULL TEXT

page 2

page 4

page 5

research
08/16/2023

B-stability of numerical integrators on Riemannian manifolds

We propose a generalization of nonlinear stability of numerical one-step...
research
12/17/2021

A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations

Deep Neural Networks are widely used for solving complex problems in sev...
research
01/22/2019

Fast and Robust Shortest Paths on Manifolds Learned from Data

We propose a fast, simple and robust algorithm for computing shortest pa...
research
08/15/2023

Riemannian geometry for efficient analysis of protein dynamics data

An increasingly common viewpoint is that protein dynamics data sets resi...
research
01/26/2021

Statistical models and probabilistic methods on Riemannian manifolds

This entry contains the core material of my habilitation thesis, soon to...
research
04/11/2019

Probabilistic Permutation Synchronization using the Riemannian Structure of the Birkhoff Polytope

We present an entirely new geometric and probabilistic approach to synch...
research
08/17/2011

Fat Triangulations and Differential Geometry

We study the differential geometric consequences of our previous result ...

Please sign up or login with your details

Forgot password? Click here to reset