Measures of entropy from data using infinitely divisible Kernels

Luis Gonzalo Sanchez Giraldo, Murali Rao, Jose C. Principe

Research output: Contribution to journalArticlepeer-review

81 Scopus citations

Abstract

Information theory provides principled ways to analyze different inference and learning problems, such as hypothesis testing, clustering, dimensionality reduction, classification, and so forth. However, the use of information theoretic quantities as test statistics, that is, as quantities obtained from empirical data, poses a challenging estimation problem that often leads to strong simplifications, such as Gaussian models, or the use of plug in density estimators that are restricted to certain representation of the data. In this paper, a framework to nonparametrically obtain measures of entropy directly from data using operators in reproducing kernel Hilbert spaces defined by infinitely divisible kernels is presented. The entropy functionals, which bear resemblance with quantum entropies, are defined on positive definite matrices and satisfy similar axioms to those of Renyi's definition of entropy. Convergence of the proposed estimators follows from concentration results on the difference between the ordered spectrum of the Gram matrices and the integral operators associated to the population quantities. In this way, capitalizing on both the axiomatic definition of entropy and on the representation power of positive definite kernels, the proposed measure of entropy avoids the estimation of the probability distribution underlying the data. Moreover, estimators of kernel-based conditional entropy and mutual information are also defined. Numerical experiments on independence tests compare favorably with state-of-the-art.

Original languageEnglish
Article number6954500
Pages (from-to)535-548
Number of pages14
JournalIEEE Transactions on Information Theory
Volume61
Issue number1
DOIs
StatePublished - Jan 1 2015

Bibliographical note

Publisher Copyright:
© 2014 IEEE.

Keywords

  • Independence test
  • Infinitely divisible kernels
  • Learning
  • Positive definite functions
  • Renyi's Entropy

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences

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