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E(n) Equivariant Message Passing Cellular Networks

Veljko Kovač, Erik J. Bekkers, Pietro Liò, Floor Eijkelboom

TL;DR

The paper introduces E(n) Equivariant Cellular Message Passing Networks (EMPCNs), a generalization of $E(n)$-equivariant GNNs to CW-complexes to enable higher-order, topological message passing with enhanced expressivity. It proposes a scalable, decoupled variant that decouples node-to-node communication from cellular higher-order lifting, preserving computational efficiency. Empirical results across N-body, QM9, and CMU Motion Capture show EMPCNs achieve near-state-of-the-art performance without steerability, with the decoupled version offering improved data-efficiency and generalization in limited-resource settings. The work highlights a scalable framework for integrating geometric invariants over complex topologies, suggesting broad applicability to structured data where rings, cycles, and other higher-order formations are informative.

Abstract

This paper introduces E(n) Equivariant Message Passing Cellular Networks (EMPCNs), an extension of E(n) Equivariant Graph Neural Networks to CW-complexes. Our approach addresses two aspects of geometric message passing networks: 1) enhancing their expressiveness by incorporating arbitrary cells, and 2) achieving this in a computationally efficient way with a decoupled EMPCNs technique. We demonstrate that EMPCNs achieve close to state-of-the-art performance on multiple tasks without the need for steerability, including many-body predictions and motion capture. Moreover, ablation studies confirm that decoupled EMPCNs exhibit stronger generalization capabilities than their non-topologically informed counterparts. These findings show that EMPCNs can be used as a scalable and expressive framework for higher-order message passing in geometric and topological graphs

E(n) Equivariant Message Passing Cellular Networks

TL;DR

The paper introduces E(n) Equivariant Cellular Message Passing Networks (EMPCNs), a generalization of -equivariant GNNs to CW-complexes to enable higher-order, topological message passing with enhanced expressivity. It proposes a scalable, decoupled variant that decouples node-to-node communication from cellular higher-order lifting, preserving computational efficiency. Empirical results across N-body, QM9, and CMU Motion Capture show EMPCNs achieve near-state-of-the-art performance without steerability, with the decoupled version offering improved data-efficiency and generalization in limited-resource settings. The work highlights a scalable framework for integrating geometric invariants over complex topologies, suggesting broad applicability to structured data where rings, cycles, and other higher-order formations are informative.

Abstract

This paper introduces E(n) Equivariant Message Passing Cellular Networks (EMPCNs), an extension of E(n) Equivariant Graph Neural Networks to CW-complexes. Our approach addresses two aspects of geometric message passing networks: 1) enhancing their expressiveness by incorporating arbitrary cells, and 2) achieving this in a computationally efficient way with a decoupled EMPCNs technique. We demonstrate that EMPCNs achieve close to state-of-the-art performance on multiple tasks without the need for steerability, including many-body predictions and motion capture. Moreover, ablation studies confirm that decoupled EMPCNs exhibit stronger generalization capabilities than their non-topologically informed counterparts. These findings show that EMPCNs can be used as a scalable and expressive framework for higher-order message passing in geometric and topological graphs
Paper Structure (31 sections, 9 equations, 4 figures, 3 tables)

This paper contains 31 sections, 9 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Symmetry-Informed Deep Learning
  4. Message Passing Neural Networks
  5. Geometric MPNNs
  6. Topological MPNNs
  7. Geometric & Topological MPNNs
  8. Methodology
  9. $\mathrm{E}(n)$ Equivariant Cellular Message Passing
  10. Invariants
  11. Barycentric Subdivision
  12. Generalized Volume and Area
  13. Other Invariants
  14. Scalable Convex Hull Estimation
  15. Geometric Weisfeiler-Lehman
  16. ...and 16 more sections

Figures (4)

  • Figure 1: Visualization of a graph (a), a simplicial complex (b), and a cellular complex (c). As observed, a simplicial complex cannot represent arbitrary polygons.
  • Figure 2: Change of invariants after position updates: a) Initial graph with arrows indicating the future displacement of the respective nodes. b) Displaced graph showing the updated positions of the nodes. c) Cell decomposition into two simplices.
  • Figure 3: Pipeline of Decoupled EMPCNs. The input graph is split into two: the cellular lifted graph for higher-order message passing and the fully connected graph for direct node communication.
  • Figure 4: Ablation study on the $\alpha$ property of QM9, comparing decoupled EMPCN (red) and EGNN (blue). Both models were trained with 100% data (solid lines) and 5% data (dashed lines), and evaluated on MAE ($\downarrow$) across different numbers of layers.