Estimating R\'enyi's $\alpha$-Cross-Entropies in a Matrix-Based Way

Isaac J. Sledge; Jose C. Principe

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arXiv · 2109.11737

Estimating R\'enyi's $\alpha$-Cross-Entropies in a Matrix-Based Way

Isaac J. Sledge·Jose C. Principe

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Abstract

Conventional information-theoretic quantities assume access to probability distributions. Estimating such distributions is not trivial. Here, we consider function-based formulations of cross entropy that sidesteps this a priori estimation requirement. We propose three measures of R\'enyi's $\alpha$-cross-entropies in the setting of reproducing-kernel Hilbert spaces. Each measure has its appeals. We prove that we can estimate these measures in an unbiased, non-parametric, and minimax-optimal way. We do this via sample-constructed Gram matrices. This yields matrix-based estimators of R\'enyi's $\alpha$-cross-entropies. These estimators satisfy all of the axioms that R\'enyi established for divergences. Our cross-entropies can thus be used for assessing distributional differences. They are also appropriate for handling high-dimensional distributions, since the convergence rate of our estimator is independent of the sample dimensionality. Python code for implementing these measures can be found at https://github.com/isledge/MBRCE

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BibTeX RIS

Isaac J. Sledge, Jose C. Principe. 2021-09-24. Estimating R\'enyi's $\alpha$-Cross-Entropies in a Matrix-Based Way. https://arxiv.org/abs/2109.11737

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