GMR-Conv: An Efficient Rotation and Reflection Equivariant Convolution Kernel Using Gaussian Mixture Rings

TL;DR

GMR-Conv introduces Gaussian Mixture Ring kernels to achieve rotation and reflection equivariance with high efficiency, addressing discretization errors that plague traditional radial symmetric convolutions. The method decomposes the full kernel into Gaussian-smoothed rings and a channel-wise 1-by-1 mixing, delivering large receptive fields with reduced parameters and favorable computational complexity $O(HW n(k^2+C_{in}C_{out}))$. The authors provide continuous-space equivariance proofs, extensive 2D and 3D experiments across eight classification tasks and segmentation, and ablations showing when smoothing and family of rings improve performance. Overall, GMR-Conv offers a plug-and-play, hardware-friendly approach that preserves geometric priors while achieving competitive or superior performance on orientation-agnostic data, highlighting a practical path toward robust equivariant networks.

Abstract

Symmetry, where certain features remain invariant under geometric transformations, can often serve as a powerful prior in designing convolutional neural networks (CNNs). While conventional CNNs inherently support translational equivariance, extending this property to rotation and reflection has proven challenging, often forcing a compromise between equivariance, efficiency, and information loss. In this work, we introduce Gaussian Mixture Ring Convolution (GMR-Conv), an efficient convolution kernel that smooths radial symmetry using a mixture of Gaussian-weighted rings. This design mitigates discretization errors of circular kernels, thereby preserving robust rotation and reflection equivariance without incurring computational overhead. We further optimize both the space and speed efficiency of GMR-Conv via a novel parameterization and computation strategy, allowing larger kernels at an acceptable cost. Extensive experiments on eight classification and one segmentation datasets demonstrate that GMR-Conv not only matches conventional CNNs' performance but can also surpass it in applications with orientation-less data. GMR-Conv is also proven to be more robust and efficient than the state-of-the-art equivariant learning methods. Our work provides inspiring empirical evidence that carefully applied radial symmetry can alleviate the challenges of information loss, marking a promising advance in equivariant network architectures. The code is available at https://github.com/XYPB/GMR-Conv.

Figures (17)

• Figure 1: Advantages of GMR-Conv. (a) High performance with low complexity: Model size, GPU memory, and performance on rotated CIFAR-10 test set. The size of each circle reflects the model size. (b) Efficient computation: Time of 1,000 random convolution operations at different convolutional kernel widths. (c) Superior equivariance: Per-degree performance on orientation-less NCT-CRC dataset of pathological images. The SoTA method is underlined, and ours is highlighted in bold.
• Figure 2: Building the GMR-Conv Kernel Efficiently. (a) We build the GMR-Conv kernel from the Gaussian rings controlled by $\Theta_\sigma$ and weighted by $\Theta_w$. The final kernel will be a weighted sum of each ring. (b) To convolve the proposed GMR-Conv kernel efficiently, we split it into a per-Gaussian ring depthwise convolution and a $\Theta_w$ weighted 1-by-1 convolution to reduce computational complexity.
• Figure 3: Classification Accuracy Across Rotation Angles. We visualize the classification accuracy (%) per test rotation angles and the box plot of per-angle accuracy for (a) CIFAR-10 krizhevsky2009learning, (b) NWPU-10 su2019object, (c) MTARSI wu2020benchmark, (d) NCT-CRC kather2019predicting, and (e) Patch-Camelyon bejnordi2017diagnostic. We scale the radial axis to better visualize differences in performance. We omit the E(2) baseline with only $0^{th}$-order filters as its performance is much lower than the E(2)-WRN16. Symm-R18 serves as the naive radial symmetry convolution baseline.
• Figure 4: Model Parameter Efficiency Evaluation. We plot the averaged accuracy on rotated CIFAR-10 test sets for each model, varying its channel size and number of parameters.
• Figure S2: CIFAR-10 krizhevsky2009learning Kernel Design Experiment Across Rotation Degree. We plot the detailed accuracy curve for each rotation angle in a radar plot for each group of kernel design experiments (\ref{['tab:arch']}). CK. refers to the circular kernel, GMR refers to the Gaussian Mixture Ring, and Sigma is the parameter $\Theta_\sigma$ that controls each Gaussian function's standard deviation. Our choice of model design shows the best performance.