In plain words: Instead of letting nodes repeatedly swap information with neighbors, this method builds each shape's representation directly from its coordinates using a math tool that tracks how quantities change across space. It beat the usual neighbor-swapping networks on geometric graphs whose node features are coordinates.
Abstract · Simplicial Representation Learning with Neural $k$-Forms
Geometric deep learning extends deep learning to incorporate information about the geometry and topology data, especially in complex domains like graphs. Despite the popularity of message passing in this field, it has limitations such as the need for graph rewiring, ambiguity in interpreting data, and over-smoothing. In this paper, we take a different approach, focusing on leveraging geometric information from simplicial complexes embedded in $\mathbb{R}^n$ using node coordinates. We use differential k-forms in \mathbb{R}^n to create representations of simplices, offering interpretability and geometric consistency without message passing. This approach also enables us to apply differential geometry tools and achieve universal approximation. Our method is efficient, versatile, and applicable to various input complexes, including graphs, simplicial complexes, and cell complexes. It outperforms existing message passing neural networks in harnessing information from geometrical graphs with node features serving as coordinates.
Kelly Maggs, Celia Hacker, Bastian Rieck
arXiv:2312.08515 · cs.LG, math.AT · submitted Dec 13, 2023 · updated Mar 15, 2024
abstract · pdf · html · Accepted at ICLR 2024 (https://openreview.net/forum?id=Djw0XhjHZb)