Measures of Entropy from Data Using Infinitely Divisible Kernels

Information theory provides principled models to analyze different inference and learning problems such as hypothesis testing, clustering, dimensionality reduction, classification, among others. However, the use of information theoretic quantities as test statistics, that is, as quantities obtained from empirical data, posses 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 non-parametrically obtain measures of entropy directly from data using infinitely divisible kernels is presented. In resemblance to quantum information theory, functionals on positive definite matrices that satisfy similar properties to the ones given in Renyi's axiomatic definition of entropy are defined. Therefore, the estimation of the probability law underlying the data is avoided, capitalizing on the representation power that positive definite kernels bring. In the proposed framework, analogues to quantities such as conditional entropy and mutual information are obtained. Numerical validation using the proposed quantities to test independence is provided. In the considered examples, the proposed framework can achieve state of the art performances.