Families of Optimal Transport Kernels for Cell Complexes
This work addresses a lack of machine learning methods for cell complexes, providing tools for comparing and analyzing these structures, though it appears incremental as it extends existing optimal transport concepts to a new domain.
The paper tackles the problem of learning on cell complexes by deriving an explicit Wasserstein distance expression based on a Hodge-Laplacian matrix, resulting in a structurally meaningful measure and novel kernels for probability measures on CW complexes.
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.