Simplicial Representation Learning with Neural $k$-Forms

Kelly Maggs, Celia Hacker, and Bastian Rieck

This repository contains the code for the paper Simplicial Representation Learning with Neural k-Forms, presented at ICLR 2024, the International Conference on Learning Representations.

Abstract. 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.

Please use the following citation for our work:

@inproceedings{Maggs24a,
  title         = {Simplicial Representation Learning with Neural $k$-forms},
  author        = {Kelly Maggs and Celia Hacker and Bastian Rieck},
  year          = 2024,
  booktitle     = {International Conference on Learning Representations},
  url           = {https://openreview.net/forum?id=Djw0XhjHZb},
}

Installation with pip or poetry

With poetry install or pip install ., the base packages will be set up. However, we recommend installing torch and torch-geometric manually, since their installation procedure differs based on CPU or GPU versions.

For the CPU versions, this should work:

$ pip install torch torchvision torchaudio --index-url https://download.pytorch.org/whl/cpu
$ pip install torch_geometric
$ pip install pyg_lib torch_scatter torch_sparse torch_cluster torch_spline_conv -f https://data.pyg.org/whl/torch-2.0.0+cpu.html
$ pip install torchmetrics pytorch-lightning

Experiments and Tutorials

Please check out several guided tutorial notebooks in the notebooks folder. The following notebooks are available:

• Synthetic path classification experiments
• Synthetic surface classification
• Visualising eigenvectors of the simplicial $1$-Laplacian operator

We have also integrated our neural $k$-forms into a simple architecture for graph classification. To run these experiments, reproducing Table 1 and Table 2 from the main paper, run the graphs.py script or module with an appropriate name:

$ python -m neural_k_forms.graphs --name BZR                # To run our model
$ python -m neural_k_forms.graphs --name BZR --baseline GCN # To run a baseline

Architecture

Most of the heavy lifting is done in chains.py. As outlined in the paper, the main ingredient is the calculation of the integration matrix, which is done by the generate_integration_matrix(). Currently, this is realised for $1$-forms (vector fields) only, please refer to the tutorials to see examples of how to run this in higher dimensions.