Entropy-based closure for probabilistic learning on manifolds

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

1 MSME - Laboratoire de Modélisation et Simulation Multi Echelle
2 USC - University of Southern California
3 Sandia National Laboratories [Livermore]
4 LLNL - Lawrence Livermore National Laboratory

In a recent paper, the authors proposed a general methodology for probabilistic learning on man-ifolds. 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 mani-fold structure is learned using diffusion manifolds and the statistical sample generation is accomplished using a projected Itô 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 ε. Currently, ε is more or less arbitrarily chosen. In this paper, we propose a selection criterion for identifying an optimal value of ε, 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.

Mots clés: Statistical learning; Probabilistic learning; Concentration of probability; Measure concentration; Probability distribution on manifolds; Random sampling generator; MCMC generator; Diffusion maps; Smoothing parameter; Entropy principle

Domaines: Sciences de l'ingénieur [physics]; Probabilités [math.PR]; Statistiques [math.ST]

Christian Soize, Roger Ghanem, C. Safta, X. Huan, Z.P. P Vane, et al.. Entropy-based closure for probabilistic learning on manifolds. Journal of Computational Physics, 2019, 388, pp.518-533. ⟨10.1016/j.jcp.2018.12.029⟩. ⟨hal-02100250⟩