# Entropy-based closure for probabilistic learning on manifolds

arXiv: https://arxiv.org/abs/1803.08161 · HTML: https://mlsanity.com/paper/1803.08161/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Mar 2018.

## Built on by (most cited first, 6 of 6)

- [Uncertainty Quantification in Machine Learning for Engineering Design and Health Prognostics: A Tutorial](https://mlsanity.com/paper/2305.04933/) (2023)
- [Generative Learning of Densities on Manifolds](https://mlsanity.com/paper/2503.03963/) (2025)
- [Transient anisotropic kernel for probabilistic learning on manifolds](https://mlsanity.com/paper/2407.21435/) (2024)
- [Dimensionality reduction can be used as a surrogate model for high-dimensional forward uncertainty quantification](https://mlsanity.com/paper/2402.04582/) (2024)
- [Enabling Probabilistic Learning on Manifolds through Double Diffusion Maps](https://mlsanity.com/paper/2506.02254/) (2025)
- [Supervised Learning of Random Neural Architectures Structured by Latent Random Fields on Compact Boundaryless Multiply-Connected Manifolds](https://mlsanity.com/paper/2512.10407/) (2025)

## Abstract (arXiv)

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 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.