Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics

Philipp Hennig, Søren Hauberg

Research output: Chapter in Book/Report/Conference proceedingArticle in proceedingsResearchpeer-review

156 Downloads (Pure)

Abstract

We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics on Riemannian manifolds, where nonanalytic ordinary di_erential equations are involved in virtually all computations. The probabilistic formulation permits marginalising the uncertainty of the numerical solution such that statistics are less sensitive to inaccuracies. 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 speedup over the state of the art. Our approach is an argument for a wider point that uncertainty caused by numerical calculations should be tracked throughout the pipeline of machine learning algorithms.
Original languageEnglish
Title of host publicationProceedings of the 17th International Conference on Artifficial Intelligence and Statistics (AISTATS 2014)
Publication date2014
Pages347-355
Publication statusPublished - 2014
Externally publishedYes
Event17th International Conference on Artificial Intelligence and Statistics - Reykjavik, Iceland
Duration: 22 Apr 201425 Apr 2014
Conference number: 17
http://www.aistats.org/aistats2014/

Conference

Conference17th International Conference on Artificial Intelligence and Statistics
Number17
CountryIceland
CityReykjavik
Period22/04/201425/04/2014
Internet address
SeriesJMLR: Workshop and Conference Proceedings
Volume33
ISSN1938-7228

Cite this

Hennig, P., & Hauberg, S. (2014). Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics. In Proceedings of the 17th International Conference on Artifficial Intelligence and Statistics (AISTATS 2014) (pp. 347-355). JMLR: Workshop and Conference Proceedings, Vol.. 33 http://jmlr.org/proceedings/papers/v33/