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

2025-04-03

AI Summary

Existing equivariant CNNs struggle to simultaneously achieve strict SE(2) equivariance (under continuous rotations and reflections), computational efficiency, and preservation of full input information. To address this, we propose GMR-Conv, a novel convolutional kernel parameterized via Gaussian Mixture Rings (GMRs). GMR-Conv directly models radially symmetric kernels in the continuous domain using polar-coordinate parameterization and lightweight equivariant computation, thereby achieving exact SE(2) equivariance without discretization error, supporting arbitrary kernel sizes with zero precision loss. Unlike discrete-group convolutions or spectral methods, GMR-Conv breaks the traditional trade-off among equivariance, efficiency, and fidelity. Experiments across eight classification and one segmentation benchmark demonstrate that GMR-Conv matches or exceeds standard CNN performance while significantly improving robustness and inference efficiency over state-of-the-art equivariant approaches.

Technology Category

• Machine Learning: Kernel Methods
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Search and Optimization: Non-convex Optimization

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Assisted, interactive, and conversational search
• Security and Privacy: Security and privacy of machine learning and AI applications

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.

Problem

• Extending equivariance to rotation and reflection in CNNs without efficiency loss
• Mitigating discretization errors in circular kernels for robust symmetry preservation
• Balancing equivariance, computational cost, and information loss in network design

Innovation

• Gaussian Mixture Ring Convolution kernel
• Efficient rotation and reflection equivariance
• Optimized space and speed efficiency