Universal Invariant and Equivariant Graph Neural Networks (1905.04943v2)

Abstract: Graph Neural Networks (GNN) come in many flavors, but should always be either invariant (permutation of the nodes of the input graph does not affect the output) or equivariant (permutation of the input permutes the output). In this paper, we consider a specific class of invariant and equivariant networks, for which we prove new universality theorems. More precisely, we consider networks with a single hidden layer, obtained by summing channels formed by applying an equivariant linear operator, a pointwise non-linearity and either an invariant or equivariant linear operator. Recently, Maron et al. (2019) showed that by allowing higher-order tensorization inside the network, universal invariant GNNs can be obtained. As a first contribution, we propose an alternative proof of this result, which relies on the Stone-Weierstrass theorem for algebra of real-valued functions. Our main contribution is then an extension of this result to the equivariant case, which appears in many practical applications but has been less studied from a theoretical point of view. The proof relies on a new generalized Stone-Weierstrass theorem for algebra of equivariant functions, which is of independent interest. Finally, unlike many previous settings that consider a fixed number of nodes, our results show that a GNN defined by a single set of parameters can approximate uniformly well a function defined on graphs of varying size.

• The paper establishes that invariant and equivariant GNNs serve as universal approximators for continuous graph functions.
• The authors introduce a generalized Stone-Weierstrass theorem to rigorously handle the permutation symmetries in equivariant networks.
• The findings enable the design of flexible GNN architectures applicable to diverse fields such as molecular simulation and social network analysis.

Universal Invariant and Equivariant Graph Neural Networks

The paper "Universal Invariant and Equivariant Graph Neural Networks" by Nicolas Keriven and Gabriel Peyré offers theoretical advancements in the field of Graph Neural Networks (GNNs) through the exploration of invariant and equivariant networks. This work adds rigor to the understanding of when and how these networks can be considered universal approximators of continuous functions on graphs, a key property that ensures their practical applicability to a wide range of problems.

Theoretical Contributions

One of the core contributions of this paper lies in establishing universality theorems for a class of GNNs. The authors focus on networks that have a single hidden layer and are constructed using equivariant linear operators, followed by pointwise non-linearity, and terminating with either invariant or equivariant linear output layers. These constructions are crucial because they ensure that the inductive biases of the networks are aligned with the underlying symmetries of graph data.

The first significant contribution is an alternative proof of universality for invariant GNNs. This invariant case was previously explored by Maron et al., 2019, who demonstrated the universal approximation capabilities of invariant GNNs that incorporate higher-order tensorization. However, the authors of this paper provide a novel proof using the Stone-Weierstrass theorem for algebras of real-valued functions. This proof is notable for its applicability to graphs of varying sizes, whereas prior work often assumed a fixed size.

The primary advancement offered by the authors is extending these universality results to the equivariant case—a domain less explored theoretically. The proof presented necessitates the development of a generalized version of the Stone-Weierstrass theorem, tailored to equivariant functions. This extension is not trivial, as equivariant networks must retain alignment between the permutations in the input and output, reflecting a more intricate form of symmetry compared to invariant networks.

Implications and Future Directions

From a practical standpoint, the research outlines a pathway for constructing GNN architectures capable of faithfully modeling the vast diversity of graph-structured data. By demonstrating that a single set of GNN parameters can be applied to graphs of different sizes while maintaining approximation properties, the authors facilitate more flexible and broadly applicable models that fit into real-world scenarios such as molecular simulations, social network analysis, and more.

Moreover, the insights from this paper could inspire further research into deep learning models that respect and leverage symmetries inherent in different data types beyond graphs. Future work could potentially explore the trade-offs involved in the approximation power concerning the order of tensorization, as well as extensions to broader classes of transformations beyond permutations.

While this paper focuses on theoretical guarantees, it forms a valuable foundation for the design of robust GNNs. Researchers may build on these results to empirically test these architectures in various domains, further refining and tuning the practicality of the proposed models.

Overall, the theoretical groundwork laid by Keriven and Peyré helps solidify the understanding of GNNs' potential to serve as universal approximators for graph data, enriching the toolkit available for both theoretical explorations and practical applications in machine learning and artificial intelligence.