On Hölder Projective Divergences

Handle URI:
http://hdl.handle.net/10754/624040
Title:
On Hölder Projective Divergences
Authors:
Nielsen, Frank; Sun, Ke; Marchand-Maillet, Stephane
Abstract:
We describe a framework to build distances by measuring the tightness of inequalities and introduce the notion of proper statistical divergences and improper pseudo-divergences. We then consider the Holder ordinary and reverse inequalities and present two novel classes of Holder divergences and pseudo-divergences that both encapsulate the special case of the Cauchy-Schwarz divergence. We report closed-form formulas for those statistical dissimilarities when considering distributions belonging to the same exponential family provided that the natural parameter space is a cone (e.g., multivariate Gaussians) or affine (e.g., categorical distributions). Those new classes of Holder distances are invariant to rescaling and thus do not require distributions to be normalized. Finally, we show how to compute statistical Holder centroids with respect to those divergences and carry out center-based clustering toy experiments on a set of Gaussian distributions which demonstrate empirically that symmetrized Holder divergences outperform the symmetric Cauchy-Schwarz divergence.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Nielsen F, Sun K, Marchand-Maillet S (2017) On Hölder Projective Divergences. Entropy 19: 122. Available: http://dx.doi.org/10.3390/e19030122.
Publisher:
MDPI AG
Journal:
Entropy
Issue Date:
16-Mar-2017
DOI:
10.3390/e19030122
Type:
Article
ISSN:
1099-4300
Sponsors:
The authors gratefully thank the referees for their comments. Ke Sun is funded by King Abdullah University of Science and Technology (KAUST).
Additional Links:
http://www.mdpi.com/1099-4300/19/3/122
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorNielsen, Franken
dc.contributor.authorSun, Keen
dc.contributor.authorMarchand-Maillet, Stephaneen
dc.date.accessioned2017-06-05T06:02:24Z-
dc.date.available2017-06-05T06:02:24Z-
dc.date.issued2017-03-16en
dc.identifier.citationNielsen F, Sun K, Marchand-Maillet S (2017) On Hölder Projective Divergences. Entropy 19: 122. Available: http://dx.doi.org/10.3390/e19030122.en
dc.identifier.issn1099-4300en
dc.identifier.doi10.3390/e19030122en
dc.identifier.urihttp://hdl.handle.net/10754/624040-
dc.description.abstractWe describe a framework to build distances by measuring the tightness of inequalities and introduce the notion of proper statistical divergences and improper pseudo-divergences. We then consider the Holder ordinary and reverse inequalities and present two novel classes of Holder divergences and pseudo-divergences that both encapsulate the special case of the Cauchy-Schwarz divergence. We report closed-form formulas for those statistical dissimilarities when considering distributions belonging to the same exponential family provided that the natural parameter space is a cone (e.g., multivariate Gaussians) or affine (e.g., categorical distributions). Those new classes of Holder distances are invariant to rescaling and thus do not require distributions to be normalized. Finally, we show how to compute statistical Holder centroids with respect to those divergences and carry out center-based clustering toy experiments on a set of Gaussian distributions which demonstrate empirically that symmetrized Holder divergences outperform the symmetric Cauchy-Schwarz divergence.en
dc.description.sponsorshipThe authors gratefully thank the referees for their comments. Ke Sun is funded by King Abdullah University of Science and Technology (KAUST).en
dc.publisherMDPI AGen
dc.relation.urlhttp://www.mdpi.com/1099-4300/19/3/122en
dc.rightsThis is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).en
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/en
dc.subjectHolder inequalitiesen
dc.subjectHolder divergencesen
dc.subjectprojective divergencesen
dc.subjectCauchy-Schwarz divergenceen
dc.subjectHolder escort divergencesen
dc.subjectskew Bhattacharyya divergencesen
dc.subjectexponential familiesen
dc.subjectconic exponential familiesen
dc.subjectescort distributionen
dc.subjectclusteringen
dc.titleOn Hölder Projective Divergencesen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalEntropyen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionComputer Science Department LIX, École Polytechnique, 91128 Palaiseau Cedex, Franceen
dc.contributor.institutionSony Computer Science Laboratories Inc., Tokyo 141-0022, Japanen
dc.contributor.institutionComputer Vision and Multimedia Laboratory (Viper), University of Geneva, CH-1211 Geneva, Switzerlanden
kaust.authorSun, Keen
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