Group Equivariant Convolution (GConv) can effectively handle rotational symmetry data. They assume uniform and strict rotational symmetry across all features, as the transformations under the specific group. However, real-world data rarely conforms to strict rotational symmetry commonly referred to as Rotational Symmetry-Breaking in the system or dataset, making GConv unable to adapt effectively to this phenomenon. Motivated by this, we propose a simple but highly effective method to address this problem, which utilizes a set of learnable biases called the $G$-Biases under the group order to break strict group constraints and achieve \textbf{R}elaxed \textbf{R}otational \textbf{E}quivarant \textbf{Conv}olution (RREConv). We conduct extensive experiments to validate Relaxed Rotational Equivariance on rotational symmetry groups $\mathcal{C}_n$ (e.g. $\mathcal{C}_2$, $\mathcal{C}_4$, and $\mathcal{C}_6$ groups). Further experiments demonstrate that our proposed RREConv-based methods achieve excellent performance, compared to existing GConv-based methods in classification and detection tasks on natural image datasets.

本研究解决了当前组等变卷积（GConv）在处理旋转对称数据时无法适应实际数据中常见的旋转对称破裂问题。我们提出了一种名为$G$-偏差的可学习偏置，突破严格的组约束，从而实现了松弛旋转等变卷积（RREConv）。实验表明，基于RREConv的方法在分类和检测任务中表现优异，优于现有的GConv方法。

通过$G$-偏差实现松弛旋转等变性