  In a recent paper, the authors proposed a general methodology for
probabilistic learning on manifolds. The method was used to generate numerical
samples that are statistically consistent with an existing dataset construed as
a realization from a non-Gaussian random vector. The manifold structure is
learned using diffusion manifolds and the statistical sample generation is
accomplished using a projected Ito stochastic differential equation. This
probabilistic learning approach has been extended to polynomial chaos
representation of databases on manifolds and to probabilistic nonconvex
constrained optimization with a fixed budget of function evaluations. The
methodology introduces an isotropic-diffusion kernel with hyperparameter
{\epsilon}. Currently, {\epsilon} is more or less arbitrarily chosen. In this
paper, we propose a selection criterion for identifying an optimal value of
{\epsilon}, based on a maximum entropy argument. The result is a comprehensive,
closed, probabilistic model for characterizing data sets with hidden
constraints. This entropy argument ensures that out of all possible models,
this is the one that is the most uncertain beyond any specified constraints,
which is selected. Applications are presented for several databases.

Probability;Machine Learning