Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Products (2401.10216v2)

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.

• The paper presents the Gaunt Tensor Product to reduce computational complexity in equivariant operations from cubic to linear.
• It leverages Fast Fourier Transforms in a 2D Fourier basis to enable rapid tensor product computations for efficient feature interactions.
• Experimental validation shows enhanced speed and accuracy in equivariant convolutions and many-body interaction models across diverse architectures.

Understanding Efficiency in Equivariant Neural Networks with the Gaunt Tensor Product

Introduction to Equivariant Neural Networks

Equivariant neural networks are a key advancement in machine learning-related 3D data modeling. These networks are designed to consistently transform their outputs in response to changes in their inputs, maintaining symmetries that naturally occur in data such as molecular structures or 3D images. A typical way to implement equivariance in neural networks involves using tensor products of irreducible representations (irreps), employing tools from representation theory. While powerful, this approach faces computational challenges as operations become complex, especially when dealing with higher-order tensors.

Advancing Computational Efficiency with the Gaunt Tensor Product

A team of researchers has proposed an innovative solution to the computational obstacles posed by conventional methods when tensor products of irreps are used for enforcing equivariance. They introduce the Gaunt Tensor Product, aiming to provide a significant computational speed-up by exploiting the mathematical relationship between the widely used Clebsch-Gordan coefficients and the Gaunt coefficients.

The key insight of this work is that the aforementioned coefficients, central to equivariant operations, can be connected to efficiently computable integrals involving spherical harmonics. By transitioning to a 2D Fourier basis representation, the challenging tensor products can then be rapidly computed through efficient Fast Fourier Transforms (FFTs), leveraging the convolution theorem.

This novel method substantially reduces computational complexity from a curve with a steep incline (cubic in the max degree of irreps) to a more tractable linear growth curve, thereby facilitating the construction of more efficient equivariant operations across different model architectures without sacrificing their capabilities.

Application Across Various Model Architectures

The Gaunt Tensor Product is not only faster but also versatile. The authors delve into three primary classes of operations where their approach can be applied:

• Equivariant Feature Interactions: It enables features from different dimensional spaces and objects to interact, a fundamental aspect in many advanced models.
• Equivariant Convolutions: By incorporating insights from advanced models and adding sparsity, their method achieves even greater accelerations for key building blocks of equivariant message passing.
• Equivariant Many-Body Interactions: Essential for capturing complex interaction effects in molecular systems, their method utilizes a divide-and-conquer strategy to efficiently parallelize computations, making it especially effective for models predicting dynamic systems like force fields.

Experimental Validation and Impacts

Extensive experiments confirm the effectiveness and efficiency of the Gaunt Tensor Product. The approach demonstrates significant speed improvements compared to existing libraries and methods on standard benchmark datasets while maintaining or exceeding modeling accuracy. Through these experiments, the team establishes the Gaunt Tensor Product as a formidable tool for the future of equivariant neural networks, especially in fields requiring large-scale 3D molecular modeling.

The innovative approach promises to ease the computational demands imposed on researchers and pave the way for new exploration opportunities in developing advanced equivariant models. As the field of AI continues to grow, such improvements in computational methods are crucial to harness the full potential of machine learning for understanding and predicting complex physical phenomena.