## This content will become publicly available on March 27, 2020

## 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 a non-Gaussian random vector. The manifold structure is learned using diffusion manifolds and the statistical sample generation is accomplished using a projected Itˆo 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 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.

- Authors:

- Univ. Paris, Marne-La-Vallee (France)
- Univ. of Southern California, Los Angeles, CA (United States)
- Sandia National Lab. (SNL-CA), Livermore, CA (United States)
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA); Defense Advanced Research Projects Agency (DARPA)

- OSTI Identifier:
- 1526161

- Alternate Identifier(s):
- OSTI ID: 1529282

- Report Number(s):
- LLNL-JRNL-744191; SAND-2018-0462J

Journal ID: ISSN 0021-9991; 899147

- Grant/Contract Number:
- AC52-07NA27344; AC04-94AL85000

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 388; Journal Issue: C; Journal ID: ISSN 0021-9991

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Soize, C., Ghanem, R., Safta, C., Huan, X., Vane, Z. P., Oefelein, J., Lacaze, G., Najm, H. N., Tang, Q., and Chen, X. Entropy-based closure for probabilistic learning on manifolds. United States: N. p., 2019.
Web. doi:10.1016/j.jcp.2018.12.029.
```

```
Soize, C., Ghanem, R., Safta, C., Huan, X., Vane, Z. P., Oefelein, J., Lacaze, G., Najm, H. N., Tang, Q., & Chen, X. Entropy-based closure for probabilistic learning on manifolds. United States. doi:10.1016/j.jcp.2018.12.029.
```

```
Soize, C., Ghanem, R., Safta, C., Huan, X., Vane, Z. P., Oefelein, J., Lacaze, G., Najm, H. N., Tang, Q., and Chen, X. Wed .
"Entropy-based closure for probabilistic learning on manifolds". United States. doi:10.1016/j.jcp.2018.12.029.
```

```
@article{osti_1526161,
```

title = {Entropy-based closure for probabilistic learning on manifolds},

author = {Soize, C. and Ghanem, R. and Safta, C. and Huan, X. and Vane, Z. P. and Oefelein, J. and Lacaze, G. and Najm, H. N. and Tang, Q. and Chen, X.},

abstractNote = {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 Itˆo 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 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.},

doi = {10.1016/j.jcp.2018.12.029},

journal = {Journal of Computational Physics},

number = C,

volume = 388,

place = {United States},

year = {2019},

month = {3}

}