  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
