Evolution equations with anisotropic distributions and diffusion PCA

Stefan Horst Sommer

doi:10.1007/978-3-319-25040-3_1

Evolution equations with anisotropic distributions and diffusion PCA

Stefan Horst Sommer

Department of Computer Science

4 Citations (Scopus)

Abstract

This paper presents derivations of evolution equations for the family of paths that in the Diffusion PCA framework are used for approximating data likelihood. The paths that are formally interpreted as most probable paths generalize geodesics in extremizing an energy functional on the space of differentiable curves on a manifold with connection. We discuss how the paths arise as projections of geodesics for a (non bracket-generating) sub-Riemannian metric on the frame bundle. Evolution equations in coordinates for both metric and cometric formulations of the sub-Riemannian geometry are derived. We furthermore show how rank-deficient metrics can be mixed with an underlying Riemannian metric, and we use the construction to show how the evolution equations can be implemented on finite dimensional LDDMM landmark manifolds.

Original language	English
Title of host publication	Geometric science of information : Second International Conference, GSI 2015, Palaiseau, France, October 28–30, 2015, Proceedings
Editors	Frank Nielsen, Frédéric Barbaresco
Number of pages	9
Publisher	Springer Science+Business Media
Publication date	2015
Pages	3-11
ISBN (Print)	978-3-319-25039-7
ISBN (Electronic)	978-3-319-25040-3
DOIs	https://doi.org/10.1007/978-3-319-25040-3_1
Publication status	Published - 2015
Event	International Conference, GSI 2015 - Palaiseau, France Duration: 28 Oct 2015 → 30 Oct 2015 Conference number: 2

Conference

Conference	International Conference, GSI 2015
Number	2
Country/Territory	France
City	Palaiseau
Period	28/10/2015 → 30/10/2015

Series	Lecture notes in computer science
Volume	9389
ISSN	0302-9743

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10.1007/978-3-319-25040-3_1

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Evolution equations with anisotropic distributions and diffusion PCA. / Sommer, Stefan Horst.
Geometric science of information: Second International Conference, GSI 2015, Palaiseau, France, October 28–30, 2015, Proceedings. ed. / Frank Nielsen; Frédéric Barbaresco. Springer Science+Business Media, 2015. p. 3-11 (Lecture notes in computer science, Vol. 9389).

Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review

Sommer, SH 2015, Evolution equations with anisotropic distributions and diffusion PCA. in F Nielsen & F Barbaresco (eds), Geometric science of information: Second International Conference, GSI 2015, Palaiseau, France, October 28–30, 2015, Proceedings. Springer Science+Business Media, Lecture notes in computer science, vol. 9389, pp. 3-11, International Conference, GSI 2015, Palaiseau, France, 28/10/2015. https://doi.org/10.1007/978-3-319-25040-3_1

Sommer SH. Evolution equations with anisotropic distributions and diffusion PCA. In Nielsen F, Barbaresco F, editors, Geometric science of information: Second International Conference, GSI 2015, Palaiseau, France, October 28–30, 2015, Proceedings. Springer Science+Business Media. 2015. p. 3-11. (Lecture notes in computer science, Vol. 9389). doi: 10.1007/978-3-319-25040-3_1

Sommer, Stefan Horst. / Evolution equations with anisotropic distributions and diffusion PCA. Geometric science of information: Second International Conference, GSI 2015, Palaiseau, France, October 28–30, 2015, Proceedings. editor / Frank Nielsen ; Frédéric Barbaresco. Springer Science+Business Media, 2015. pp. 3-11 (Lecture notes in computer science, Vol. 9389).

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Evolution equations with anisotropic distributions and diffusion PCA

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