High-Rank Irreducible Cartesian Tensor Decomposition and Bases of Equivariant Spaces (2412.18263v6)

Abstract: Irreducible Cartesian tensors (ICTs) play a crucial role in the design of equivariant graph neural networks, as well as in theoretical chemistry and chemical physics. Meanwhile, the design space of available linear operations on tensors that preserve symmetry presents a significant challenge. The ICT decomposition and a basis of this equivariant space are difficult to obtain for high-rank tensors. After decades of research, Bonvicini (2024) recently achieves an explicit ICT decomposition for $n=5$ with factorial time/space complexity. In this work we, for the first time, obtains decomposition matrices for ICTs up to rank $n=9$ with reduced and affordable complexity, by constructing what we call path matrices. The path matrices are obtained via performing chain-like contractions with Clebsch-Gordan matrices following the parentage scheme. We prove and leverage that the concatenation of path matrices is an orthonormal change-of-basis matrix between the Cartesian tensor product space and the spherical direct sum spaces. Furthermore, we identify a complete orthogonal basis for the equivariant space, rather than a spanning set (Pearce-Crump, 2023), through this path matrices technique. To the best of our knowledge, this is also the first analytic, rather than numerical, method for theoretically obtaining arbitrary rank orthogonal ICT decomposition matrices and orthogonal equivariant bases. We further extend our result to the arbitrary tensor product and direct sum spaces, enabling free design between different spaces while keeping symmetry. The Python code is available at https://github.com/ShihaoShao-GH/ICT-decomposition-and-equivariant-bases, where the $n=6,\dots,9$ ICT decomposition matrices are obtained in 1s, 3s, 11s, and 4m32s on 28-cores Intel(R) Xeon(R) Gold 6330 CPU @ 2.00GHz, respectively.

• The paper derives decomposition matrices for high-rank irreducible Cartesian tensors (up to n=9) using novel path matrices, significantly reducing complexity.
• This work constructs a complete orthogonal basis for the equivariant space, enabling efficient processing and achieving rapid computation times even for high tensor ranks.
• The method expands the design space for Equivariant Graph Neural Networks and has profound implications for theoretical chemistry and physics by enabling effective use of high-rank tensors.

Exploring the Design Space of Equivariant Graph Neural Networks

In the design and deployment of Equivariant Graph Neural Networks (EGNNs), irreducible Cartesian tensors (ICTs) serve as fundamental components, immensely applicable in fields like theoretical chemistry and chemical physics. However, a significant challenge in EGNNs is the difficulty in obtaining ICT decomposition and a basis for the equivariant space for high-order tensors. The paper "Free the Design Space of Equivariant Graph Neural Networks: High-Rank Irreducible Cartesian Tensor Decomposition and Bases of Equivariant Spaces" offers innovative solutions to this challenge.

Overview

The core contribution of the paper is the derivation of decomposition matrices for ICTs up to rank $n = 9$, accomplished with significantly reduced complexity compared to prior approaches. This advancement is achieved through the novel construction of "path matrices," which are generated via sequential contraction with Clebsch-Gordan matrices, following a parentage scheme. The path matrices enable an orthonormal change of basis between Cartesian tensor product spaces and spherical direct sum spaces.

Unlike preceding methodologies, which either relied on explicit factorial complexity or only managed to automate procedures without complexity reduction, this work constructs a complete orthogonal basis for the equivariant space, rather than just a spanning set. It allows for an unrestricted design between spaces of differing symmetry while maintaining computational efficiency. The authors demonstrate that the path matrices they propose offer an orthonormal basis for the equivariant space, validating the efficacy and utility of their approach.

Numerical Results and Claims

The strong numerical results presented in the paper underscore the efficiency of the new method. Decomposition matrices for ranks $n=6$ to $n=9$ are generated rapidly, with computational times of under 0.1 seconds for $n=6$ and around 4.5 minutes for $n=9$. This demonstrates a marked improvement over earlier rank-5 decompositions. Additionally, the orthogonal property of these decompositions is established theoretically, providing robustness to the solutions provided.

Practical and Theoretical Implications

Practically, this work enables EGNNs to process high-rank ICTs effectively and design equivariant layers between diverse spaces, broadening the architectural possibilities within neural networks significantly. The theoretical implications are equally profound, as the method not only complements existing EGNN frameworks but extends their capability to operate with more complex many-body interactions using high-rank tensors.

Moreover, the findings have valuable implications beyond EGNNs. By benefiting communities focused on theoretical analysis within high-rank ICTs, this research presents a bridge between machine learning advancements and theoretical disciplines such as chemistry and physics.

Future Directions

This work sets the stage for subsequent advances in both EGNN architectures and theoretical chemistry. Future developments might focus on further extending the mathematical framework to more general groups beyond $O(3)$, allowing applicability to even broader areas of symmetry-constrained learning and physical modeling. The theoretical background established for ICT decomposition could also inspire innovations in other domains where high-dimensional symmetric representation is critical.

Conclusion

This paper successfully addresses a significant bottleneck in the design and application of equivariant neural networks by providing a robust, efficient method to generate high-rank irreducible Cartesian tensor decompositions. In doing so, it enhances the theoretical and practical toolkit available to researchers working on EGNNs and related fields, positioning this method as a cornerstone for subsequent explorations and applications in AI, chemistry, and physics.