Flash EQ-Linear: Accelerating Equivariant Linear Layers via Group-wise Discrete Fourier Transform

Abstract

Equivariant networks embed geometric symmetries as structural priors through weight sharing, achieving remarkable parameter efficiency across vision tasks. However, this parameter efficiency does not translate into compute efficiency: most existing implementations unroll the structured weights into dense matrices and dispatch them to generic dense kernels, so an equivariant layer costs no fewer MACs than its non-equivariant counterpart. In this paper, we observe that the equivariant linear (EQ-Linear) layer---the most fundamental and frequently used module in modern equivariant architectures---is essentially a circular convolution along the group dimension composed with a linear transform along the channel dimension. Building on this observation, we propose Flash EQ-Linear, an exact acceleration algorithm that reduces the cost to $2(T-1)/T^2$ of the original dense formulation ($T$ is the equivariant group size) by combining the Fourier convolution theorem along the group dimension with the conjugate symmetry of the real DFT. To translate these computational savings into wall-clock speedups, we further develop dedicated CUDA kernels for the $\mathrm{p}4$ group. At the operator level, Flash EQ-Linear achieves up to $2.1\times$ forward speedup over PyTorch's highly optimized F.linear; at the network level, Flash EQ-ViT achieves up to ${1.7\times}$ end-to-end speedup over both equivariant and non-equivariant baselines. As an operator-level acceleration algorithm, Flash EQ-Linear provides plug-and-play acceleration for diverse pretrained equivariant models, including EQ-ViT, EQ-Swin, EQ-VMamba, and EQ-INR, without retraining or architectural changes. Code is available at https://github.com/zhongchenzhao/FlashEQLinear.