[Paper Review] Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces

Yinshuang Xu, Jiahui Lei+2 more|arXiv (Cornell University)|Jun 16, 2022

Advanced Neuroimaging Techniques and Applications4 citations

TL;DR

This paper introduces a unified Fourier-based framework for designing group-equivariant convolutional networks on homogeneous spaces, leveraging spectral sparsity in Fourier coefficients when the stabilizer subgroup is a compact Lie group. It proposes a novel equivariant nonlinearity via lifting to the group, applying elementwise nonlinearity, and projecting back, achieving state-of-the-art performance in spherical vector field regression, point cloud classification, and molecular completion on SO(3) and SE(3).

ABSTRACT

We introduce a unified framework for group equivariant networks on homogeneous spaces derived from a Fourier perspective. We consider tensor-valued feature fields, before and after a convolutional layer. We present a unified derivation of kernels via the Fourier domain by leveraging the sparsity of Fourier coefficients of the lifted feature fields. The sparsity emerges when the stabilizer subgroup of the homogeneous space is a compact Lie group. We further introduce a nonlinear activation, via an elementwise nonlinearity on the regular representation after lifting and projecting back to the field through an equivariant convolution. We show that other methods treating features as the Fourier coefficients in the stabilizer subgroup are special cases of our activation. Experiments on $SO(3)$ and $SE(3)$ show state-of-the-art performance in spherical vector field regression, point cloud classification, and molecular completion.

Motivation & Objective

• To unify the design of equivariant kernels and nonlinearities for group convolutional networks on homogeneous spaces using a Fourier-domain perspective.
• To exploit spectral sparsity in Fourier coefficients of lifted feature fields when the stabilizer subgroup is a compact Lie group, enabling efficient kernel design.
• To develop a general, equivariant nonlinear activation by lifting fields to the group, applying elementwise nonlinearity, and projecting back via convolution.
• To demonstrate state-of-the-art performance across multiple equivariance benchmarks, including 3D shape classification and molecular structure completion.
• To unify and generalize existing approaches—such as norm-based and Clebsch-Gordan-based nonlinearities—under a single Fourier-based framework.

Proposed method

• Lift tensor-valued feature fields from the homogeneous space to Mackey functions on the acting group, exploiting the group structure for equivariance.
• Utilize the Fourier transform of Mackey functions to reveal block-sparse patterns in the Fourier domain when the stabilizer is a compact Lie group.
• Design equivariant convolutional kernels as sparse matrices in the Fourier domain, avoiding explicit algebraic or numerical constraints.
• Implement a nonlinear activation by lifting the field to the group, applying an elementwise nonlinearity (e.g., ReLU), and projecting back via an equivariant convolution.
• Ensure equivariance by construction through the use of group-invariant projections and the regular representation in the Fourier domain.
• Apply the framework to SO(3) and SE(3), using bandwidths and projection schemes to maintain computational efficiency and equivariance.

Experimental results

Research questions

• RQ1 Can spectral sparsity in the Fourier domain of lifted feature fields on homogeneous spaces be exploited to unify kernel and nonlinearity design in equivariant networks?
• RQ2 Does a general, Fourier-based framework exist that subsumes existing methods like norm-based or Clebsch-Gordan-based nonlinearities for equivariant networks?
• RQ3 Can the proposed kernel design via Fourier sparsity achieve state-of-the-art performance in tasks requiring equivariance to SO(3) and SE(3)?
• RQ4 How does the proposed nonlinearity compare to existing approaches in terms of expressivity and performance on vector field and point cloud tasks?
• RQ5 To what extent does the framework generalize across different homogeneous spaces and stabilizer groups, particularly when the stabilizer is a compact Lie group?

Key findings

• The Fourier coefficients of Mackey functions on the group are block-sparse when the stabilizer subgroup is a compact Lie group, enabling efficient and unified kernel design.
• The proposed nonlinearity generalizes prior methods: the nonlinearity in Poulenard & Guibas (2021) is a special case of the proposed lifting-and-projecting framework.
• On the ModelNet40 dataset, the method achieves 89.7% accuracy under SO(3) augmentation, matching or exceeding state-of-the-art models including TFN[mlp]-P and VN-DGCNN.
• In molecular structure completion, the model achieves 98.0% accuracy and 0.06Å MAE on 19-atom molecules, outperforming TFN (93.9% accuracy, 0.14Å MAE).
• The framework enables state-of-the-art performance in spherical vector field regression and 3D point cloud classification, demonstrating strong generalization across tasks.
• The method maintains equivariance while enabling higher expressivity through the proposed nonlinearity, as validated in ablation and benchmarking experiments.