Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics

Philipp Hennig, Søren Hauberg
Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics, PMLR 33:347-355, 2014.

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 non-analytic ordinary differential 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 speed-up 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v33-hennig14, title = {{Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics}}, author = {Hennig, Philipp and Hauberg, Søren}, booktitle = {Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics}, pages = {347--355}, year = {2014}, editor = {Kaski, Samuel and Corander, Jukka}, volume = {33}, series = {Proceedings of Machine Learning Research}, address = {Reykjavik, Iceland}, month = {22--25 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v33/hennig14.pdf}, url = {https://proceedings.mlr.press/v33/hennig14.html}, 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 non-analytic ordinary differential 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 speed-up 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.} }
Endnote
%0 Conference Paper %T Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics %A Philipp Hennig %A Søren Hauberg %B Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2014 %E Samuel Kaski %E Jukka Corander %F pmlr-v33-hennig14 %I PMLR %P 347--355 %U https://proceedings.mlr.press/v33/hennig14.html %V 33 %X 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 non-analytic ordinary differential 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 speed-up 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.
RIS
TY - CPAPER TI - Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics AU - Philipp Hennig AU - Søren Hauberg BT - Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics DA - 2014/04/02 ED - Samuel Kaski ED - Jukka Corander ID - pmlr-v33-hennig14 PB - PMLR DP - Proceedings of Machine Learning Research VL - 33 SP - 347 EP - 355 L1 - http://proceedings.mlr.press/v33/hennig14.pdf UR - https://proceedings.mlr.press/v33/hennig14.html AB - 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 non-analytic ordinary differential 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 speed-up 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. ER -
APA
Hennig, P. & Hauberg, S.. (2014). Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics. Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 33:347-355 Available from https://proceedings.mlr.press/v33/hennig14.html.

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