On geodesic exponential kernels
Research output: Chapter in Book/Report/Conference proceeding › Conference abstract in proceedings › Research › peer-review
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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
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TY - ABST
T1 - On geodesic exponential kernels
AU - Feragen, Aasa
AU - Lauze, François
AU - Hauberg, Søren
PY - 2015
Y1 - 2015
N2 - 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.
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.
U2 - 10.1007/978-3-319-24261-3
DO - 10.1007/978-3-319-24261-3
M3 - Conference abstract in proceedings
AN - SCOPUS:84959214477
SN - 9781467369640
SP - 211
EP - 213
BT - Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition
A2 - Feragen, Aasa
A2 - Pelillo, Marcello
A2 - Loog, Marco
PB - Springer
T2 - 3rd International Workshop on Similarity-Based Pattern Recognition, SIMBAD 2015
Y2 - 12 October 2015 through 14 October 2015
ER -
ID: 160634529