Group Equivariant CNNs (G-CNNs) have shown promising efficacy in various tasks, owing to their ability to capture hierarchical features in an equivariant manner. However, their equivariance is fixed to the symmetry of the whole group, limiting adaptability to diverse partial symmetries in real-world datasets, such as limited rotation symmetry of handwritten digit images and limited color-shift symmetry of flower images. Recent efforts address this limitation, one example being Partial G-CNN which restricts the output group space of convolution layers to break full equivariance. However, such an approach still fails to adjust equivariance levels across data. In this paper, we propose a novel approach, Variational Partial G-CNN (VP G-CNN), to capture varying levels of partial equivariance specific to each data instance. VP G-CNN redesigns the distribution of the output group elements to be conditioned on input data, leveraging variational inference to avoid overfitting. This enables the model to adjust its equivariance levels according to the needs of individual data points. Additionally, we address training instability inherent in discrete group equivariance models by redesigning the reparametrizable distribution. We demonstrate the effectiveness of VP G-CNN on both toy and real-world datasets, including MNIST67-180, CIFAR10, ColorMNIST, and Flowers102. Our results show robust performance, even in uncertainty metrics.

翻译：群等变卷积神经网络（G-CNN）因其能够以等变方式捕捉层次化特征，已在多种任务中展现出良好的性能。然而，其等变性固定于整个群的对称性，难以适应现实数据集中多样的部分对称性，例如手写数字图像的有限旋转对称性和花卉图像的有限颜色偏移对称性。近期研究尝试解决这一局限，例如部分G-CNN通过限制卷积层的输出群空间来打破完全等变性。但此类方法仍无法根据数据动态调整等变程度。本文提出一种新颖方法——变分部分G-CNN（VP G-CNN），以捕捉针对每个数据实例的不同部分等变程度。VP G-CNN通过变分推理重构输出群元素的分布，使其以输入数据为条件，从而避免过拟合。这使得模型能够根据个体数据点的需求自适应调整等变程度。此外，我们通过重新设计可重参数化分布，解决了离散群等变模型中固有的训练不稳定问题。我们在合成数据集和真实数据集（包括MNIST67-180、CIFAR10、ColorMNIST和Flowers102）上验证了VP G-CNN的有效性。实验结果表明，该方法即使在不确定性度量方面也表现出稳健的性能。