Invariant and Equivariant Graph Networks (1812.09902v2)

Abstract: Invariant and equivariant networks have been successfully used for learning images, sets, point clouds, and graphs. A basic challenge in developing such networks is finding the maximal collection of invariant and equivariant linear layers. Although this question is answered for the first three examples (for popular transformations, at-least), a full characterization of invariant and equivariant linear layers for graphs is not known. In this paper we provide a characterization of all permutation invariant and equivariant linear layers for (hyper-)graph data, and show that their dimension, in case of edge-value graph data, is 2 and 15, respectively. More generally, for graph data defined on k-tuples of nodes, the dimension is the k-th and 2k-th Bell numbers. Orthogonal bases for the layers are computed, including generalization to multi-graph data. The constant number of basis elements and their characteristics allow successfully applying the networks to different size graphs. From the theoretical point of view, our results generalize and unify recent advancement in equivariant deep learning. In particular, we show that our model is capable of approximating any message passing neural network Applying these new linear layers in a simple deep neural network framework is shown to achieve comparable results to state-of-the-art and to have better expressivity than previous invariant and equivariant bases.

• The paper introduces a framework characterizing permutation invariant and equivariant linear layers for graphs, specifying invariant (2) and equivariant (15) dimensions for edge-value data.
• It computes orthogonal bases for these layers, ensuring consistent network performance across graphs of varying sizes and facilitating broader applications.
• The study demonstrates that the proposed layers can approximate any message passing neural network, offering enhanced expressivity and practical implementation benefits.

Invariant and Equivariant Graph Networks

The paper "Invariant and Equivariant Graph Networks" offers significant advancements in the theoretical understanding and practical application of invariant and equivariant networks, particularly for graph data. Invariant and equivariant models have been extensively utilized in learning various data types, including images, sets, point clouds, and graphs. However, the challenge of characterizing maximal invariant and equivariant linear layers for graphs has remained unresolved until now. This paper presents a comprehensive framework for permutation invariant and equivariant linear layers for graph data, providing both theoretical insights and practical implementations.

Key Findings and Contributions

The authors deliver a detailed characterization of permutation invariant and equivariant linear layers specifically for (hyper-)graphs. For edge-value graph data, they identify that the dimension of invariant linear layers is 2, and for equivariant linear layers, it is 15. These dimensions are expressed more generally in terms of the $k$-th and $2k$-th Bell numbers, representing graph data on $k$-tuples of nodes.

Orthogonal bases for these layers are computed, enabling the application of networks to graphs of varying sizes. The constant number of basis elements, independent of node size, enhances this applicability. Importantly, the research generalizes recent advances in equivariant deep learning, demonstrating that the model can approximate any message passing neural network, thus suggesting comparable expressivity to existing bases.

Numerical Results and Claims

The authors provide compelling numerical evidence to support their claims. Applying their new linear layers within a simple neural network framework, they achieve results comparable to state-of-the-art alternatives. This is particularly evident in tasks involving different graph sizes, where the invariant and equivariant linear layers constructed using their bases demonstrate better expressivity than previous models.

Implications and Future Directions

The implications of this research are extensive both practically and theoretically. On the practical side, the ability to construct efficient and expressive neural networks for graph data without size restriction promotes broader application in various domains requiring graph representations. Theoretically, the work advances the understanding of symmetry in learning models, potentially guiding the development of more generalized frameworks for other data structures.

Looking forward, this research lays a foundation for future work in expanding equivariant models to multi-graph and multi-set data scenarios. Moreover, the established connection to message passing neural networks opens up pathways to explore deeper integration with existing graph-based learning methodologies, perhaps leading to novel architectures with enhanced capabilities.

The paper represents a significant step forward in the understanding and application of invariant and equivariant networks, setting a solid groundwork for further exploration and innovation in the field.