# Probabilistic Learning on Manifolds (PLoM) with Partition

arXiv: https://arxiv.org/abs/2102.10894 · HTML: https://mlsanity.com/paper/2102.10894/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Feb 2021.

## Built on by (most cited first, 8 of 8)

- [Generative Learning of Densities on Manifolds](https://mlsanity.com/paper/2503.03963/) (2025)
- [On the Correspondence between Gaussian Processes and Geometric Harmonics](https://mlsanity.com/paper/2110.02296/) (2021)
- [Transient anisotropic kernel for probabilistic learning on manifolds](https://mlsanity.com/paper/2407.21435/) (2024)
- [Probabilistic learning inference of boundary value problem with uncertainties based on Kullback-Leibler divergence under implicit constraints](https://mlsanity.com/paper/2202.05112/) (2022)
- [Probabilistic learning constrained by realizations using a weak formulation of Fourier transform of probability measures](https://mlsanity.com/paper/2205.03078/) (2022)
- [Conformal Disentanglement and Latent-Space Curation: A Neural Framework for Perspective Synthesis, Differentiation and Targeted Generation](https://mlsanity.com/paper/2408.15344/) (2024)
- [Application of probabilistic modeling and automated machine learning framework for high-dimensional stress field](https://mlsanity.com/paper/2303.16869/) (2023)
- [Enabling Probabilistic Learning on Manifolds through Double Diffusion Maps](https://mlsanity.com/paper/2506.02254/) (2025)

## Abstract (arXiv)

The probabilistic learning on manifolds (PLoM) introduced in 2016 has solved difficult supervised problems for the ``small data'' limit where the number N of points in the training set is small. Many extensions have since been proposed, making it possible to deal with increasingly complex cases. However, the performance limit has been observed and explained for applications for which N is very small (50 for example) and for which the dimension of the diffusion-map basis is close to N. For these cases, we propose a novel extension based on the introduction of a partition in independent random vectors. We take advantage of this novel development to present improvements of the PLoM such as a simplified algorithm for constructing the diffusion-map basis and a new mathematical result for quantifying the concentration of the probability measure in terms of a probability upper bound. The analysis of the efficiency of this novel extension is presented through two applications.