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

Datalogisk Institut

35 Citationer (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.

Originalsprog	Engelsk
Titel	Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition
Redaktører	Aasa Feragen, Marcello Pelillo, Marco Loog
Antal sider	3
Forlag	Springer
Publikationsdato	2015
Sider	211-213
ISBN (Trykt)	9781467369640
DOI	https://doi.org/10.1007/978-3-319-24261-3
Status	Udgivet - 2015
Begivenhed	3rd International Workshop on Similarity-Based Pattern Recognition, SIMBAD 2015 - Copenhagen, Danmark Varighed: 12 okt. 2015 → 14 okt. 2015

Konference

Konference	3rd International Workshop on Similarity-Based Pattern Recognition, SIMBAD 2015
Land/Område	Danmark
By	Copenhagen
Periode	12/10/2015 → 14/10/2015

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10.1007/978-3-319-24261-3

Citationsformater

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

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

Publikation: Bidrag til bog/antologi/rapport › Konferenceabstrakt i proceedings › Forskning › peer review

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

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doi = "10.1007/978-3-319-24261-3",

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AU - Hauberg, Søren

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AB - 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.

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On geodesic exponential kernels

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