Simplicial Representation Learning with Neural $k$-Forms
This work addresses the problem of geometric representation learning for researchers and practitioners in domains like graphs and simplicial complexes, offering an incremental improvement over prior methods.
The paper tackles the limitations of message passing in geometric deep learning by proposing a method that uses differential k-forms to create representations of simplices from node coordinates, achieving improved performance over existing message passing neural networks on geometrical graphs.
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.