标题： GMR-Conv：一种使用高斯混合环的高效旋转和反射等变卷积核 显示英文标题

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

摘要： 对称性，即某些特征在几何变换下保持不变，常常可以用作设计卷积神经网络（CNN）的强大先验。尽管传统的CNN天然支持平移等变性，但将其扩展到旋转和反射等变性一直是个挑战，通常需要在等变性、效率和信息损失之间做出妥协。本文中，我们引入了高斯混合环卷积（GMR-Conv），这是一种高效的卷积核，利用加权高斯环的混合来平滑径向对称性。这种设计减少了圆形核的离散化误差，从而在不增加计算开销的情况下保持了鲁棒的旋转和反射等变性。我们还通过一种新颖的参数化和计算策略优化了GMR-Conv的空间和速度效率，使得在可接受的成本下可以使用更大的核。在八个分类数据集和一个分割数据集上的广泛实验表明，GMR-Conv不仅能够匹配传统CNN的性能，还可以在无方向数据的应用中超越它们。此外，GMR-Conv被证明比最先进的等变学习方法更鲁棒和高效。我们的工作提供了令人信服的经验证据，表明谨慎应用径向对称性可以缓解信息丢失的挑战，标志着等变网络架构的一个有希望的进步。代码可在https://github.com/XYPB/GMR-Conv获取。 显示英文摘要

摘要： 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.