Entropy-based closure for probabilistic learning on manifolds

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 as 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. Currently, is more or less arbitrarily chosen. In this paper, we propose a selection criterion for identifying an optimal value of, based on an entropy argument. The result is a comprehensive, closed, probabilistic model for characterizing data sets with hidden constraints. Applications are presented for several databases.

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Citation
C. Soize, R. Ghanem, C. Safta, X. Huan, Z. P. Vane, J. C. Oefelein, G. Lacaze, H. N. Najm, Q. Tang, and X. Chen. Entropy-based closure for probabilistic learning on manifolds. Submitted, 2017.

BibTeX

@article{Soize2017,
author = {Soize, Christian and Ghanem, Roger G. and Safta, Cosmin and Huan, Xun and Vane, Zachary P. and Oefelein, Joseph C. and Lacaze, Guilhem and Najm, Habib N. and Tang, Qi. and Chen, Xiao},
journal = {Submitted},
title = {{Entropy-based closure for probabilistic learning on manifolds}},
year = {2017}
}