Entropy-based closure for probabilistic learning on manifolds

TL;DR

This paper introduces an entropy-based criterion for selecting the diffusion kernel parameter in probabilistic learning on manifolds, leading to a comprehensive model that captures data uncertainty and hidden constraints.

Contribution

It proposes a maximum entropy-based method for choosing the diffusion parameter, enhancing probabilistic models on manifolds with a principled selection criterion.

Findings

Optimal epsilon improves data modeling accuracy

Entropy-based selection captures hidden data constraints

Method applied successfully to multiple datasets

Abstract

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.