Open Problem: Kernel methods on manifolds and metric spaces. What is the probability of a positive definite geodesic exponential kernel?

Aasa Feragen, Søren Hauberg
29th Annual Conference on Learning Theory, PMLR 49:1647-1650, 2016.

Abstract

Radial kernels are well-suited for machine learning over general geodesic metric spaces, where pairwise distances are often the only computable quantity available. We have recently shown that geodesic exponential kernels are only positive definite for all bandwidths when the input space has strong linear properties. This negative result hints that radial kernel are perhaps not suitable over geodesic metric spaces after all. Here, however, we present evidence that large intervals of bandwidths exist where geodesic exponential kernels have high probability of being positive definite over finite datasets, while still having significant predictive power. From this we formulate conjectures on the probability of a positive definite kernel matrix for a finite random sample, depending on the geometry of the data space and the spread of the sample.

Cite this Paper

BibTeX
@InProceedings{pmlr-v49-feragen16, title = {Open Problem: Kernel methods on manifolds and metric spaces. What is the probability of a positive definite geodesic exponential kernel?}, author = {Feragen, Aasa and Hauberg, Søren}, booktitle = {29th Annual Conference on Learning Theory}, pages = {1647--1650}, year = {2016}, editor = {Feldman, Vitaly and Rakhlin, Alexander and Shamir, Ohad}, volume = {49}, series = {Proceedings of Machine Learning Research}, address = {Columbia University, New York, New York, USA}, month = {23--26 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v49/feragen16.pdf}, url = {https://proceedings.mlr.press/v49/feragen16.html}, abstract = {Radial kernels are well-suited for machine learning over general geodesic metric spaces, where pairwise distances are often the only computable quantity available. We have recently shown that geodesic exponential kernels are only positive definite for all bandwidths when the input space has strong linear properties. This negative result hints that radial kernel are perhaps not suitable over geodesic metric spaces after all. Here, however, we present evidence that large intervals of bandwidths exist where geodesic exponential kernels have high probability of being positive definite over finite datasets, while still having significant predictive power. From this we formulate conjectures on the probability of a positive definite kernel matrix for a finite random sample, depending on the geometry of the data space and the spread of the sample.} }
Endnote
%0 Conference Paper %T Open Problem: Kernel methods on manifolds and metric spaces. What is the probability of a positive definite geodesic exponential kernel? %A Aasa Feragen %A Søren Hauberg %B 29th Annual Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2016 %E Vitaly Feldman %E Alexander Rakhlin %E Ohad Shamir %F pmlr-v49-feragen16 %I PMLR %P 1647--1650 %U https://proceedings.mlr.press/v49/feragen16.html %V 49 %X Radial kernels are well-suited for machine learning over general geodesic metric spaces, where pairwise distances are often the only computable quantity available. We have recently shown that geodesic exponential kernels are only positive definite for all bandwidths when the input space has strong linear properties. This negative result hints that radial kernel are perhaps not suitable over geodesic metric spaces after all. Here, however, we present evidence that large intervals of bandwidths exist where geodesic exponential kernels have high probability of being positive definite over finite datasets, while still having significant predictive power. From this we formulate conjectures on the probability of a positive definite kernel matrix for a finite random sample, depending on the geometry of the data space and the spread of the sample.
RIS
TY - CPAPER TI - Open Problem: Kernel methods on manifolds and metric spaces. What is the probability of a positive definite geodesic exponential kernel? AU - Aasa Feragen AU - Søren Hauberg BT - 29th Annual Conference on Learning Theory DA - 2016/06/06 ED - Vitaly Feldman ED - Alexander Rakhlin ED - Ohad Shamir ID - pmlr-v49-feragen16 PB - PMLR DP - Proceedings of Machine Learning Research VL - 49 SP - 1647 EP - 1650 L1 - http://proceedings.mlr.press/v49/feragen16.pdf UR - https://proceedings.mlr.press/v49/feragen16.html AB - Radial kernels are well-suited for machine learning over general geodesic metric spaces, where pairwise distances are often the only computable quantity available. We have recently shown that geodesic exponential kernels are only positive definite for all bandwidths when the input space has strong linear properties. This negative result hints that radial kernel are perhaps not suitable over geodesic metric spaces after all. Here, however, we present evidence that large intervals of bandwidths exist where geodesic exponential kernels have high probability of being positive definite over finite datasets, while still having significant predictive power. From this we formulate conjectures on the probability of a positive definite kernel matrix for a finite random sample, depending on the geometry of the data space and the spread of the sample. ER -
APA
Feragen, A. & Hauberg, S.. (2016). Open Problem: Kernel methods on manifolds and metric spaces. What is the probability of a positive definite geodesic exponential kernel?. 29th Annual Conference on Learning Theory, in Proceedings of Machine Learning Research 49:1647-1650 Available from https://proceedings.mlr.press/v49/feragen16.html.

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