Data driven Dirichlet sampling on manifolds

TL;DR

This paper introduces a new sampling method on manifolds using Dirichlet distribution, enabling efficient and accurate data generation respecting the manifold structure, with applications in neural networks, uncertainty analysis, and engineering.

Contribution

The paper proposes a novel Dirichlet-based sampling technique specifically designed for manifolds, improving sampling efficiency and accuracy over existing methods.

Findings

Effective sampling on various manifolds demonstrated

Method reduces computational effort significantly

Successful application to engineering problem with gas seals

Abstract

This article presents a novel method to sampling on manifolds based on the Dirichlet distribution. The proposed strategy allows to completely respect the underlying manifold around which data is observed, and to do massive samplings with low computational effort. This can be very helpful, for instance, in neural networks training process, as well as in uncertainty analysis and stochastic optimization. Due to its simplicity and efficiency, we believe that the new method has great potential. Three manifolds (two dimensional ring, Mobius strip and spider geometry) are considered to test the proposed methodology, and then it is employed to an engineering application, related to gas seal coefficients.