On geodesic exponential kernels

Aasa Feragen; François Lauze; Søren Hauberg

doi:10.1007/978-3-319-24261-3

On geodesic exponential kernels

Aasa Feragen, François Lauze, Søren Hauberg

Department of Computer Science

35 Citations (Scopus)

Abstract

We consider kernel methods on general geodesic metric spaces and provide both negative and positive results. First we show that the common Gaussian kernel can only be generalized to a positive definite kernel on a geodesic metric space if the space is flat. As a result, for data on a Riemannian manifold, the geodesic Gaussian kernel is only positive definite if the Riemannian manifold is Euclidean. This implies that any attempt to design geodesic Gaussian kernels on curved Riemannian manifolds is futile. However, we show that for spaces with conditionally negative definite distances the geodesic Laplacian kernel can be generalized while retaining positive definiteness. This implies that geodesic Laplacian kernels can be generalized to some curved spaces, including spheres and hyperbolic spaces. Our theoretical results are verified empirically.

Original language	English
Title of host publication	Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition
Editors	Aasa Feragen, Marcello Pelillo, Marco Loog
Number of pages	3
Publisher	Springer
Publication date	2015
Pages	211-213
ISBN (Print)	9781467369640
DOIs	https://doi.org/10.1007/978-3-319-24261-3
Publication status	Published - 2015
Event	3rd International Workshop on Similarity-Based Pattern Recognition, SIMBAD 2015 - Copenhagen, Denmark Duration: 12 Oct 2015 → 14 Oct 2015

Conference

Conference	3rd International Workshop on Similarity-Based Pattern Recognition, SIMBAD 2015
Country/Territory	Denmark
City	Copenhagen
Period	12/10/2015 → 14/10/2015

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10.1007/978-3-319-24261-3

Cite this

On geodesic exponential kernels. / Feragen, Aasa; Lauze, François; Hauberg, Søren.

Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition. ed. / Aasa Feragen; Marcello Pelillo; Marco Loog. Springer, 2015. p. 211-213.

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

Feragen, A, Lauze, F & Hauberg, S 2015, On geodesic exponential kernels. in A Feragen, M Pelillo & M Loog (eds), Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition. Springer, pp. 211-213, 3rd International Workshop on Similarity-Based Pattern Recognition, SIMBAD 2015, Copenhagen, Denmark, 12/10/2015. https://doi.org/10.1007/978-3-319-24261-3

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ER -

On geodesic exponential kernels

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