Relaxed Rotational Equivariance via $G$-Biases in Vision

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.

SBDet: A Symmetry-Breaking Object Detector via Relaxed Rotation-Equivariance

Introducing Group Equivariant Convolution (GConv) empowers models to explore symmetries hidden in visual data, improving their performance. However, in real-world scenarios, objects or scenes often exhibit perturbations of a symmetric system, specifically a deviation from a symmetric architecture, which can be characterized by a non-trivial action of a symmetry group, known as Symmetry-Breaking. Traditional GConv methods are limited by the strict operation rules in the group space, only ensuring features remain strictly equivariant under limited group transformations, making it difficult to adapt to Symmetry-Breaking or non-rigid transformations. Motivated by this, we introduce a novel Relaxed Rotation GConv (R2GConv) with our defined Relaxed Rotation-Equivariant group $\mathbf{R}_4$. Furthermore, we propose a Relaxed Rotation-Equivariant Network (R2Net) as the backbone and further develop the Symmetry-Breaking Object Detector (SBDet) for 2D object detection built upon it. Experiments demonstrate the effectiveness of our proposed R2GConv in natural image classification tasks, and SBDet achieves excellent performance in object detection tasks with improved generalization capabilities and robustness.