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

Department of Computer Science

1 Citation (Scopus)

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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.

Original language	English
Title of host publication	29th Annual Conference on Learning Theory
Editors	Vitaly Feldman, Alexander Rakhlin, Ohad Shamir
Number of pages	4
Publication date	6 Jun 2016
Pages	1647–1650
Publication status	Published - 6 Jun 2016
Event	29th Conference on Learning Theory - New York, United States Duration: 23 Jun 2016 → 26 Jun 2016 Conference number: 29

Conference

Conference	29th Conference on Learning Theory
Number	29
Country/Territory	United States
City	New York
Period	23/06/2016 → 26/06/2016

Series	JMLR: Workshop and Conference Proceedings
Volume	49

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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? In V. Feldman, A. Rakhlin, & O. Shamir (Eds.), 29th Annual Conference on Learning Theory (pp. 1647–1650). JMLR: Workshop and Conference Proceedings Vol. 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. ed. / Vitaly Feldman; Alexander Rakhlin; Ohad Shamir. 2016. p. 1647–1650 (JMLR: Workshop and Conference Proceedings, Vol. 49).

Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › 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? in V Feldman, A Rakhlin & O Shamir (eds), 29th Annual Conference on Learning Theory. JMLR: Workshop and Conference Proceedings, vol. 49, pp. 1647–1650, 29th Conference on Learning Theory, New York, United States, 23/06/2016. <http://www.jmlr.org/proceedings/papers/v49/feragen16.html>

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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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