Probabilistic solutions to differential equations and their application to riemannian statistics

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 learning algorithms.