Open problem: kernel methods on manifolds and metric spaces: what is the probability of a positive definite geodesic exponential kernel?

Aasa Feragen; Søren Hauberg

Open problem: kernel methods on manifolds and metric spaces: what is the probability of a positive definite geodesic exponential kernel?

Aasa Feragen, Søren Hauberg

Datalogisk Institut

1 Citationer (Scopus)

88 Downloads (Pure)

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.

Originalsprog	Engelsk
Titel	29th Annual Conference on Learning Theory
Redaktører	Vitaly Feldman, Alexander Rakhlin, Ohad Shamir
Antal sider	4
Publikationsdato	6 jun. 2016
Sider	1647–1650
Status	Udgivet - 6 jun. 2016
Begivenhed	29th Conference on Learning Theory - New York, USA Varighed: 23 jun. 2016 → 26 jun. 2016 Konferencens nummer: 29

Konference

Konference	29th Conference on Learning Theory
Nummer	29
Land/Område	USA
By	New York
Periode	23/06/2016 → 26/06/2016

Navn	JMLR: Workshop and Conference Proceedings
Vol/bind	49

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Feragen_2016_Open_problem_kernel_methodsForlagets udgivne version, 273 KBLicens: Ikke-specificeret

http://www.jmlr.org/proceedings/papers/v49/feragen16.htmlLicens: Ikke-specificeret

Citationsformater

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? I V. Feldman, A. Rakhlin, & O. Shamir (red.), 29th Annual Conference on Learning Theory (s. 1647–1650). JMLR: Workshop and Conference Proceedings Bind 49 http://www.jmlr.org/proceedings/papers/v49/feragen16.html

Open problem: kernel methods on manifolds and metric spaces : what is the probability of a positive definite geodesic exponential kernel? / Feragen, Aasa; Hauberg, Søren.

29th Annual Conference on Learning Theory. red. / Vitaly Feldman; Alexander Rakhlin; Ohad Shamir. 2016. s. 1647–1650 (JMLR: Workshop and Conference Proceedings, Bind 49).

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

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? i V Feldman, A Rakhlin & O Shamir (red), 29th Annual Conference on Learning Theory. JMLR: Workshop and Conference Proceedings, bind 49, s. 1647–1650, 29th Conference on Learning Theory, New York, USA, 23/06/2016. <http://www.jmlr.org/proceedings/papers/v49/feragen16.html>

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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.",

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Open problem: kernel methods on manifolds and metric spaces: what is the probability of a positive definite geodesic exponential kernel?

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