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Conventional information-theoretic quantities assume access to probabilitydistributions. Estimating such distributions is not trivial. Here, we considerfunction-based formulations of cross entropy that sidesteps this a prioriestimation 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 inan unbiased, non-parametric, and minimax-optimal way. We do this viasample-constructed Gram matrices. This yields matrix-based estimators ofR\'enyi's $\alpha$-cross-entropies. These estimators satisfy all of the axiomsthat R\'enyi established for divergences. Our cross-entropies can thus be usedfor assessing distributional differences. They are also appropriate forhandling high-dimensional distributions, since the convergence rate of ourestimator is independent of the sample dimensionality.  Python code for implementing these measures can be found athttps://github.com/isledge/MBRCE

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