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Simplicial Representation Learning with Neural $k$-Forms

Kelly Maggs, Celia Hacker, Bastian Rieck

TL;DR

Geometric deep learning has leaned heavily on message passing, which can cause over-smoothing and poor long-range inference. The paper proposes neural $k$-forms, learning differentiable $k$-forms on $\ eals^n$ and integrating them over embedded $k$-simplices to produce integration matrices $X_\phi(\beta, \omega^\psi)$ that serve as geometry-aware representations without message passing. It proves a universal approximation property for neural $k$-forms in the space $\Omega^k_c(\mathbb{R}^n)$ and demonstrates practical efficiency and applicability to graphs, simplicial complexes, and cell complexes, with competitive or superior performance on graph benchmarks using far fewer parameters. The framework offers interpretable learned vector fields via 1-forms and outlines a concrete scaffold for integrating differential geometry into ML pipelines, signaling a shift toward geometry-first representations in geometric deep learning.

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.

Simplicial Representation Learning with Neural $k$-Forms

TL;DR

Geometric deep learning has leaned heavily on message passing, which can cause over-smoothing and poor long-range inference. The paper proposes neural -forms, learning differentiable -forms on and integrating them over embedded -simplices to produce integration matrices that serve as geometry-aware representations without message passing. It proves a universal approximation property for neural -forms in the space and demonstrates practical efficiency and applicability to graphs, simplicial complexes, and cell complexes, with competitive or superior performance on graph benchmarks using far fewer parameters. The framework offers interpretable learned vector fields via 1-forms and outlines a concrete scaffold for integrating differential geometry into ML pipelines, signaling a shift toward geometry-first representations in geometric deep learning.

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 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.
Paper Structure (59 sections, 5 theorems, 53 equations, 8 figures, 2 tables, 2 algorithms)

This paper contains 59 sections, 5 theorems, 53 equations, 8 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Organisation of Paper.
  3. Related Work.
  4. Background
  5. Abstract Simplicial Complexes.
  6. Affine Embeddings.
  7. Chains and Cochains over $\mathbb{R}$.
  8. Differential Forms.
  9. Scaling Functions.
  10. Integration.
  11. Neural k-Forms and Integration Matrices
  12. Neural $k$-forms.
  13. Universal Approximation.
  14. Cochain Matrices.
  15. Integration Matrices.
  16. ...and 44 more sections

Key Result

Theorem 3

Let $\alpha \in C(\mathbb{R}, \mathbb{R})$ be a non-polynomial activation function. For every $n \in \mathbb{N}$ and compactly supported $k$-form $\eta \in \Omega^k_c(\mathbb{R}^n)$ and $\epsilon > 0$ there exists a neural $k$-form $\omega^\psi$ with one hidden layer such that $\lVert \omega^\psi -

Figures (8)

  • Figure 1: An integration matrix with data in dimension $1$, where embedded oriented $1$-simplices correspond to paths and $1$-forms are canonically identified with vector fields (see \ref{['diff_form_appendix']} for details). Integration of a $1$-form against a path corresponds to path integration against the respective vector field. Thus, integration of the paired paths and $1$-forms in the left matrix recovers real values with the signs given in the right matrix.
  • Figure 2: A schematic of our proposed neural $k$-form learning architecture.
  • Figure 3: Synthetic Path Classification via Learnable $1$-forms. The $1$-form (a vector field in this case) adjusts itself to the data, resulting in distinct path representations.
  • Figure 4: Synthetic surface classification via learnable $2$-forms.
  • Figure 5: A convolutional $1$-form network with the learned convolutional filters forms.
  • ...and 3 more figures

Theorems & Definitions (12)

  • Definition 1
  • Remark 2
  • Theorem 3
  • Definition 4
  • Remark 5: Interpretability
  • Proposition 6: Multi-linearity
  • Corollary 7: Equivariance
  • Remark 8
  • Proposition 9: Lee03, 16.21
  • Theorem 10: Universal Approximation Theorem, Thm 3.1 Pinkus99
  • ...and 2 more