Probabilistic Learning on Manifolds (PLoM) with Partition

TL;DR

This paper introduces a novel extension to the Probabilistic Learning on Manifolds (PLoM) method, using partitioning to improve performance in small data scenarios with high-dimensional diffusion maps.

Contribution

The paper proposes a new partition-based extension to PLoM, simplifying basis construction and providing a mathematical measure for probability concentration.

Findings

Improved diffusion-map basis construction algorithm.

New mathematical results for probability measure concentration.

Validated effectiveness through two applications.

Abstract

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.