Entropy-based closure for probabilistic learning on manifolds

JOURNAL OF COMPUTATIONAL PHYSICS

卷:388

Article

Soize, C.¹  Ghanem, R.²  Safta, C.³  Huan, X.³  Vane, Z. P.³  Oefelein, J.³  Lacaze, G.³  Najm, H. N.³  Tang, Q.⁴  Chen, X.⁴ 

[1] Univ Paris Est Marne la Vallee, MSME UMR 8208 CNRS, 5 Bd Descartes, F-77454 Marne La Vallee, France

[2] Univ Southern Calif, 210 KAP Hall, Los Angeles, CA 90089 USA

[3] Sandia Natl Labs, 7011 East Ave, Livermore, CA 99551 USA

[4] Lawrence Livermore Natl Lab, Livermore, CA USA

关键词: Statistical learning;    Probabilistic learning;    Probability distribution on manifolds;    MCMC generator;    Diffusion maps;    Entropy principle;   

DOI  :  10.1016/j.jcp.2018.12.029

来源: Elsevier

【 摘 要 】

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. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free