Families of Optimal Transport Kernels for Cell Complexes

TL;DR

This paper introduces new kernels for comparing cell complexes using optimal transport, enabling machine learning methods to leverage both structural and feature information in CW complexes.

Contribution

It derives explicit Wasserstein distances for cell complex signals and extends the Fused Gromov-Wasserstein distance to CW complexes, introducing novel kernels for this space.

Findings

Explicit Wasserstein distance for cell complex signals derived

Extended Fused Gromov-Wasserstein distance to CW complexes

Introduced kernels for probability measures on CW complexes

Abstract

Recent advances have discussed cell complexes as ideal learning representations. However, there is a lack of available machine learning methods suitable for learning on CW complexes. In this paper, we derive an explicit expression for the Wasserstein distance between cell complex signal distributions in terms of a Hodge-Laplacian matrix. This leads to a structurally meaningful measure to compare CW complexes and define the optimal transportation map. In order to simultaneously include both feature and structure information, we extend the Fused Gromov-Wasserstein distance to CW complexes. Finally, we introduce novel kernels over the space of probability measures on CW complexes based on the dual formulation of optimal transport.