Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Products
Abstract
Developing equivariant neural networks for the E(3) group plays an important role in modeling 3D data across real-world applications. Enforcing this equivariance primarily involves the tensor products of irreducible representations (irreps). However, the computational complexity of such operations increases significantly as higher-order tensors are used. In this work, we propose a systematic approach to substantially accelerate the computation of the tensor products of irreps. We mathematically connect the commonly used Clebsch-Gordan coefficients to the Gaunt coefficients, which are integrals of products of three spherical harmonics. Through Gaunt coefficients, the tensor product of irreps becomes equivalent to the multiplication between spherical functions represented by spherical harmonics. This perspective further allows us to change the basis for the equivariant operations from spherical harmonics to a 2D Fourier basis. Consequently, the multiplication between spherical functions represented by a 2D Fourier basis can be efficiently computed via the convolution theorem and Fast Fourier Transforms. This transformation reduces the complexity of full tensor products of irreps from $\mathcal{O}(L^6)$ to $\mathcal{O}(L^3)$, where $L$ is the max degree of irreps. Leveraging this approach, we introduce the Gaunt Tensor Product, which serves as a new method to construct efficient equivariant operations across different model architectures. Our experiments on the Open Catalyst Project and 3BPA datasets demonstrate both the increased efficiency and improved performance of our approach.
AI-Generated Overview
Overview of the Paper: "Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Products"
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Research Focus: The paper presents a novel approach to accelerate the computation of tensor products of irreducible representations (irreps) in equivariant neural networks aimed at modeling 3D data, particularly focusing on the E(3) group.
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Methodology: The authors propose the "Gaunt Tensor Product," which connects Clebsch-Gordan coefficients to Gaunt coefficients, enabling the representation of tensor products as multiplications of spherical functions. The computations are then transformed into the 2D Fourier basis to exploit the convolution theorem and Fast Fourier Transforms (FFT) for enhanced efficiency.
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Results: The proposed method reduces the computational complexity of tensor products from O(L^6) to O(L^3), significantly improving both the efficiency of various equivariant operations and empirical performance on datasets such as the Open Catalyst Project and 3BPA.
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Key Contributions:
- Introduction of the Gaunt Tensor Product as a means to facilitate efficient equivariant operations across different neural network architectures.
- Establishment of a theoretical framework that connects representation theory to practical computation in neural networks, showcasing the mathematical foundation for these tensor operations.
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Significance: This work addresses a critical computational bottleneck in equivariant neural networks, allowing for higher degrees of irreps in models, which could lead to better approximation capabilities and performance in tasks involving 3D data.
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Broader Applications: The Gaunt Tensor Product could extend beyond 3D data modeling to other fields requiring efficient computation of equivariant operations, such as quantum chemistry, physics simulations, and other areas in computational graphics and machine learning where symmetry plays an integral role.