A network based approach for unbalanced optimal transport on surfaces

TL;DR

This paper introduces a neural network method for unbalanced optimal transport on surfaces represented by point clouds, overcoming mesh generation challenges and demonstrating stability and adaptability across various geometries.

Contribution

It proposes a novel neural network framework based on Hamiltonian flow for unbalanced optimal transport on point cloud surfaces, simplifying the loss function and enhancing applicability.

Findings

Method is stable with noisy point clouds

Applicable to diverse surface geometries

Simplifies the optimal transport computation

Abstract

In this paper, we present a neural network approach to address the dynamic unbalanced optimal transport problem on surfaces with point cloud representation. For surfaces with point cloud representation, traditional method is difficult to apply due to the difficulty of mesh generating. Neural network is easy to implement even for complicate geometry. Moreover, instead of solving the original dynamic formulation, we consider the Hamiltonian flow approach, i.e. Karush-Kuhn-Tucker system. Based on this approach, we can exploit mathematical structure of the optimal transport to construct the neural network and the loss function can be simplified. Extensive numerical experiments are conducted for surfaces with different geometry. We also test the method for point cloud with noise, which shows stability of this method. This method is also easy to generalize to diverse range of problems.