# Enabling Probabilistic Learning on Manifolds through Double Diffusion Maps

arXiv: https://arxiv.org/abs/2506.02254 · HTML: https://mlsanity.com/paper/2506.02254/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 45 references that are in the map.
- First posted on arXiv 2 Jun 2025.

## Builds on (most cited first, 26 of 26)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Deep Kernel Learning](https://mlsanity.com/paper/1511.02222/) (2015)
- [Permutation Invariant Graph Generation via Score-Based Generative Modeling](https://mlsanity.com/paper/2003.00638/) (2020)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [Input Warping for Bayesian Optimization of Non-stationary Functions](https://mlsanity.com/paper/1402.0929/) (2014)
- [Learning Particle Physics by Example: Location-Aware Generative Adversarial Networks for Physics Synthesis](https://mlsanity.com/paper/1701.05927/) (2017)
- [Generative adversarial networks (GAN) based efficient sampling of chemical space for inverse design of inorganic materials](https://mlsanity.com/paper/1911.05020/) (2019)
- [Double Diffusion Maps and their Latent Harmonics for Scientific Computations in Latent Space](https://mlsanity.com/paper/2204.12536/) (2022)
- [Generative Learning of the Solution of Parametric Partial Differential Equations Using Guided Diffusion Models and Virtual Observations](https://mlsanity.com/paper/2408.00157/) (2024)
- [Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets](https://mlsanity.com/paper/2010.14324/) (2020)
- [Applications of Generative Adversarial Networks in Neuroimaging and Clinical Neuroscience](https://mlsanity.com/paper/2206.07081/) (2022)
- [Probabilistic Learning on Manifolds (PLoM) with Partition](https://mlsanity.com/paper/2102.10894/) (2021)
- [Entropy-based closure for probabilistic learning on manifolds](https://mlsanity.com/paper/1803.08161/) (2018)
- [Generative Learning for Forecasting the Dynamics of Complex Systems](https://mlsanity.com/paper/2402.17157/) (2024)
- [Sampling of Bayesian posteriors with a non-Gaussian probabilistic learning on manifolds from a small dataset](https://mlsanity.com/paper/1910.12717/) (2019)
- [Micro-Macro Consistency in Multiscale Modeling: Score-Based Model Assisted Sampling of Fast/Slow Dynamical Systems](https://mlsanity.com/paper/2312.05715/) (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)
- [Generative Learning for Slow Manifolds and Bifurcation Diagrams](https://mlsanity.com/paper/2504.20375/) (2025)

## Abstract (arXiv)

We present a generative learning framework for probabilistic sampling based on an extension of the Probabilistic Learning on Manifolds (PLoM) approach, which is designed to generate statistically consistent realizations of a random vector in a finite-dimensional Euclidean space, informed by a limited (yet representative) set of observations. In its original form, PLoM constructs a reduced-order probabilistic model by combining three main components: (a) kernel density estimation to approximate the underlying probability measure, (b) Diffusion Maps to uncover the intrinsic low-dimensional manifold structure, and (c) a reduced-order Ito Stochastic Differential Equation (ISDE) to sample from the learned distribution. A key challenge arises, however, when the number of available data points N is small and the dimensionality of the diffusion-map basis approaches N, resulting in overfitting and loss of generalization. To overcome this limitation, we propose an enabling extension that implements a synthesis of Double Diffusion Maps -- a technique capable of capturing multiscale geometric features of the data -- with Geometric Harmonics (GH), a nonparametric reconstruction method that allows smooth nonlinear interpolation in high-dimensional ambient spaces. This approach enables us to solve a full-order ISDE directly in the latent space, preserving the full dynamical complexity of the system, while leveraging its reduced geometric representation. The effectiveness and robustness of the proposed method are illustrated through two numerical studies: one based on data generated from two-dimensional Hermite polynomial functions and another based on high-fidelity simulations of a detonation wave in a reactive flow.