Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics

17th International Conference on Artificial Intelligence and Statistics

数学科学;计算机科学

Philipp Hennig Søren Hauberg

PID  :  123805

来源: CEUR

【 摘 要 】

We study a probabilistic numerical method for the solution of both boundary and ini tial value problems that returns a joint Gaus sian process posterior over the solution. Such methods have concrete value in the statis tics on Riemannian manifolds, where non analytic ordinary differential equations are involved in virtually all computations. The probabilistic formulation permits marginalis ing the uncertainty of the numerical solution such that statistics are less sensitive to in accuracies. This leads to new Riemannian algorithms for mean value computations and principal geodesic analysis. Marginalisation also means results can be less precise than point estimates, enabling a noticeable speed up over the state of the art. Our approach is an argument for a wider point that uncer tainty caused by numerical calculations should be tracked throughout the pipeline of machine