Deeper and Better: Mitigating Oversmoothing and Oversquashing in MPNNs via Local Riemannian Geometry
Li Sun, Zhenhao Huang, Ming Zhang, Philip S Yu
Message Passing Neural Networks (MPNNs) achieve remarkable progress in learning on graphs, but fundamentally suffer from oversmoothing and oversquashing. There has recently been a surge of interest in fixing both issues. Existing efforts primarily conduct some global treatments that can be preferable to some regions but harmful to the others, leading to the suboptimal expressiveness. In this paper, we start with revisiting oversquashing through a global measure of spectral gap $\lambda$, and prove that the increase of $\lambda$ is exposed to gradient vanishing regarding the input features,deteriorating the efficacy of  message passing. Motivated by such theoretical insights, we propose a novel local treatment where the adaptive adjustment is introduced to message passing, according to the local structures. To this end, we bridge local Riemannian geometry with MPNNs, and establish a heterogeneous boundary condition to simultaneously address oversquashing and oversmoothing. Hence, grounded on Robin condition, we design a simple yet effective GBN network with theoretical guarantees. Extensive experiments on homophilic and heterophilic graphs show the expressiveness of GBN.