Geodesic exponential kernels: When Curvature and Linearity Conflict

Aase Feragen, François Lauze, Søren Hauberg

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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 languageEnglish
Title of host publicationProceedings of the 28th IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2015)
PublisherIEEE
Publication date2015
Pages3032-3042
DOIs
Publication statusPublished - 2015
Event28th IEEE Conference on Computer Vision and Pattern Recognition - Boston, United States
Duration: 7 Jun 201512 Jun 2015
Conference number: 28

Conference

Conference28th IEEE Conference on Computer Vision and Pattern Recognition
Number28
Country/TerritoryUnited States
CityBoston
Period07/06/201512/06/2015

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