Finsler Geometry, Graph Neural Networks, and You
This work provides a theoretical foundation for extending GNNs to anisotropic operators, addressing a limitation of Laplacian-based methods for geometric deep learning.
The paper introduces a graph neural network layer that approximates the Finsler Laplacian on manifolds, proving convergence and showing that the resulting Finslerian GNNs can recover geometry underlying nonlinear diffusion equations.
Graph neural network architectures based on the graph Laplacian approximate the Laplace-Beltrami operator, thus limiting their application to isotropic operators. As a nonlinear alternative to the Laplace-Beltrami operator, we consider estimates of the Finsler Laplacian on point clouds sampled from a manifold. We prove that these discrete estimates converge to the true operator on the manifold as the number of point samples grows. Moreover, we show that this operator can be expressed as a graph neural network layer, which we use to define a family of Finslerian graph neural networks constrained to express Finsler geometry. We show that Finslerian graph neural networks recover the geometry underlying nonlinear diffusion equations in practice.