Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics

2014

Conference Paper

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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.

Author(s): Philipp Hennig and Soren Hauberg
Book Title: Proceedings of the 17th International Conference on Artificial Intelligence and Statistics
Volume: 33
Pages: 347-355
Year: 2014
Month: April

Series: JMLR: Workshop and Conference Proceedings
Editors: S Kaski and J Corander
Publisher: Microtome Publishing

Department(s): Empirical Inference, Perceiving Systems, Probabilistic Numerics
Research Project(s): Probabilistic Numerics
Bibtex Type: Conference Paper (inproceedings)
Paper Type: Conference

Address: Brookline, MA
Event Name: AISTATS 2014
Event Place: Reykjavik, Iceland
URL: http://jmlr.org/proceedings/papers/v33/hennig14.pdf

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BibTex

@inproceedings{hennig:aistats:2014,
  title = {Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics},
  author = {Hennig, Philipp and Hauberg, S{o}ren},
  booktitle = {Proceedings of the 17th International Conference on Artificial Intelligence and Statistics},
  volume = {33},
  pages = {347-355},
  series = {JMLR: Workshop and Conference Proceedings},
  editors = { S Kaski and J Corander},
  publisher = {Microtome Publishing},
  address = {Brookline, MA},
  month = apr,
  year = {2014},
  url = {http://jmlr.org/proceedings/papers/v33/hennig14.pdf},
  month_numeric = {4}
}