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

Aasa Feragen, Søren Hauberg

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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 languageEnglish
Title of host publicationJMLR: Workshop and Conference Proceedings
Number of pages4
Volume49
Publication date2016
Pages1-4
Publication statusPublished - 2016
EventAnnual Conference on Learning Theory 2016 - Columbia University, New York, United States
Duration: 23 Jun 201626 Jun 2016
Conference number: 2016

Conference

ConferenceAnnual Conference on Learning Theory 2016
Number2016
LocationColumbia University
CountryUnited States
CityNew York
Period23/06/201626/06/2016
SeriesJMLR: Workshop and Conference Proceedings
ISSN1938-7228

Keywords

  • Kernel Methods
  • Geodesic Metric Spaces
  • Geodesic Exponential Kernel
  • Positive Definiteness
  • Curvature
  • Bandwith Selection

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