R2Det: Exploring Relaxed Rotation Equivariance in 2D Object Detection

## Abstract

Abstract 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 𝐑 4 subscript 𝐑 4 \mathbf{R}_{4} bold_R start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT . 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.

## 1 Introduction

Figure: Figure 1: The construction of convolution filters with Strict and Relaxed Rotation-Equivariance (S.R.E. and R.R.E.) on the rotation group $\mathbf{C}_{4}$.
Refer to caption: x1.png

Object detection [, ] is a vital computer vision [] task and is pivotal in various domains, including autonomous driving, geosciences, and ecology.
Recent advancements in Deep Neural Networks (DNNs) [, ] have made remarkable progress.
Nevertheless, objects within natural images often exhibit rotation and scale variations, requiring DNNs to handle geometric transformations more flexibly.
For example, in auto-driving scenes, the position and relationship of targets are complex and intricate; objects and the vehicle itself can cause rotation in the scene captured by sensors, seriously affecting target detection performance [, , , ].
More formally, the problem is deemed rotation-equivariant since rotating the input should be anticipated to induce an equivalent rotation in the output.
Let $f(\cdot)$ denote the equivariance function for a transformation $g(\cdot)$; it is known that equivariant functions preserve the symmetry of their input, and we have $f(g(\cdot))=g(f(\cdot))$.
One of the currently effective ways to solve this problem is data augmentation, which aims to improve object detection performance by rotating the dataset, which leads to a considerable cost.
Another approach is to build Equivariant Neural Networks (ENNs) [], which already incorporate symmetries and can focus on the underlying physics.
Various ENNs have provided significant improvements for the object detection task.

Although using symmetry as an inductive bias in machine learning has emerged as a powerful tool, with significant conceptual and practical breakthroughs [], a wide range of learning tasks necessitates Symmetry-Breaking [].
However, ENNs cannot effectively model Symmetry-Breaking, as the requirement for equivariance is inherently too restrictive.
Note that the notion of symmetry captures the idea that an object is essentially the same after some transformation is applied to it [].
For instance, equivariant functions can not break symmetry at the level of data samples. Consider a dataset of images with rotational symmetry, where each image depicts a single object that has been rotated.
As previously discussed, specific endeavors in ENNs aim to capturing the intrinsic symmetry inherent in images by incorporating the $\mathbf{C}_{2}$ or $\mathbf{C}_{4}$ groups.
However, these symmetries are restricted to predefined, rigid group operations.
When an object experiences symmetry-breaking events, such as rotational deformations beyond the scope of the predefined group or the introduction of minor defects on its surface, a strictly ENN may encounter difficulties in accurately representing these asymmetries.
This is because strict adherence to symmetry constraints could prevent it from distinguishing between the object’s perturbed and non-perturbed states, which is crucial for specific tasks.
Some pioneer works [, , , , , ] discussed relaxation of equivariance and claimed that relaxed ENNs can model Symmetry-Breaking in multiple domains. However, there is still a significant gap between these existing relaxed ENN works and the computer vision field.

This paper proposes a novel Relaxed Rotation GConv with a learned or controlled degree of rotation deviation.
This relaxation enables the network to recognize the approximate symmetry of the object while capturing the distinct features that arise from perturbations. It facilitates the network’s ability to model Symmetry-Breaking effectively, thereby providing a more nuanced and accurate data representation.
To the best of our knowledge, we are the first to address the Symmetry-Breaking problem formally in the object detection scenario.
The main contributions of this paper are summarized as follows:

- 1.
We propose a simple yet useful Relaxed Rotation GConv (R2GConv) to tackle the trade-off between underlying symmetry-breaking and rotation-equivariance within the image dataset.
- 2.
We propose a redesigned Symmetry-Breaking Object Detection Network (SBDet). To our knowledge, we are the first to explore the symmetry-breaking situation for the object detection task.
Experimental experiments demonstrate that our method achieves better convergence and state-of-the-art performance with
fewer parameters and better robustness.

## 2 Related works

### 2.1 Rotation-Equivariant Neural Networks and Symmetry-Breaking

As a pioneering work, the concept of an equivariant network was proposed in the Group Equivariant Convolution Neural Networks (G-CNN) [].
Rotation-Equivariant convolution or full connect layer [, , , ] guarantees the rotation-equivariance of extracted features under the group operations by a higher degree of weight sharing.
Moreover, the concept of equivariant network is also applied in Graph Neural Networks (GNN) [, ], which have demonstrated their prominence in dealing with unarchitectured data, such as molecule [] and point clouds [].

Although physical laws are governed by numerous symmetries, real-world data, such as complex datasets and graphs, often deviate from strict mathematical symmetry either due to noisy or incomplete data or to inherent Symmetry-Breaking features in the underlying system [, ].
Smidt et al. [] show that the gradients of the loss function can be used to learn a Symmetry-Breaking order parameter.
Strict constraints on the weights enforce symmetry in equivariant networks.
Relaxed equivariant networks constitute another relevant approach, permitting layer weights that enable a departure from strict equivariance.
To break symmetry at the level of individual data samples,
Wang et al. [] investigate approximately equivariant networks by incorporating relaxed weight sharing in group convolutions and weight-tying in steerable CNNs, respectively, thereby achieving a bias toward not strictly preserving symmetry.
Kaba et al. [] proposes a novel methodology for constructing relaxed equivariant multilayer perceptrons, going beyond the straightforward approach of adding noise to inputs followed by using an equivariant neural network [].
Huang et al. [] tackled graph symmetry in real-world data by leveraging graph coarsening to establish approximate symmetries and proposing a bias-variance tradeoff formula based on symmetry group selection.
Xie et al. [] introduces Symmetry-Breaking parameters sampled as model inputs from a set determined solely by input and output symmetries. They further observe that breaking more symmetry than needed is sometimes beneficial.

### 2.2 2D Object Detection

With the advent of DNNs and the increasing computational power of GPUs, DNNs have been successfully applied to various computer vision tasks, including object detection [, , ].
Object detection is a fusion of object location and object classification tasks. It involves locating objects through bounding boxes and identifying their respective categories.
Among the different object detection algorithms, the YOLO framework [, , ] has stood out for its remarkable balance of speed and accuracy, enabling the rapid and reliable identification of objects in images.
Since its inception, the YOLO family has evolved through multiple iterations and various other variations, each building upon the previous versions to address limitations and enhance performance.
The newest YOLOv8 [] uses a backbone similar to YOLOv5 [] and the C2f module to combine high-level features with contextual information to improve detection accuracy. YOLOv8 [] provided five scaled versions: YOLOv8-n (nano), YOLOv8-s (small), YOLOv8-m (medium), YOLOv8-l (large), and YOLOv8-x (extra-large).
In addition to the YOLO framework, the field of object detection and image processing has developed several other notable methods, including RCNN [], SSD [], DETR [], and Transformer-based PVT [, ].

Rotation-Equivariance has recently become a strongly desired property in object detection.
ReDet [] incorporates Rotation-Equivariant networks into the detector to extract Rotation-Equivariant features, allowing for accurate orientation prediction and substantially reducing model size.
EON [] introduces a rotation-invariant prior that addresses object detection in 3D scenes, where the bounding box should be equivariant to the object’s pose, regardless of the motion of the scene.
TED [] proposed an efficient transformation-equivariant 3D detector with competitive speed, which comprises a transformation equivariant sparse convolution backbone, transformation-equivariant Bird-Eye-View pooling, and transformation-invariant voxel pooling.
DuEqNet [] improves object detection performance by constructing a dual-layer object detection network for 3D point clouds with rotational invariance and extracting local-global invariance features.
FRED [] further decouples the invariant task (object classification) from the equivariant task (object localization), achieving fully Rotation-Equivariant oriented object detection and enabling more genuine non-axis-aligned learning.
These existing equivariant object detection approaches are primarily tailored to 3D point cloud data or specific application contexts.
They also overlook scenarios involving necessitating a relaxation of equivariant constraints, i.e., Symmetry-Breaking.
This paper further explores a more general 2D object detection in natural images.

## 3 Method

In the section, we first provide the mathematical definitions of Strict and Relaxed Rotation-Equivariance and briefly review and summarize existing Strict and Relaxed Rotation-Equivariant ENNs, which serve as preliminary knowledge.
Then, we describe the main idea of the Relaxed Rotation-Equivariant GConv module.
Finally, the Relaxed Rotation-Equivariant object detection network, Symmetry-Breaking Detection Network (SBDet), is introduced and analyzed.

### 3.1 Preliminary

Definition 1 (Strict Equivariance). A learning function $\Phi_{\textit{strict}}:\mathcal{X}\rightarrow\mathcal{Y}$ that sends elements from input space $\mathcal{X}$ to output space $\mathcal{Y}$ satisfies Strict Equivariance to a group $G$ if $\forall g,\mathbf{x}\in G\times\mathcal{X}$ there exists $\rho^{\mathcal{X}}:G\rightarrow GL\left(\mathcal{X}\right)$ and $\rho^{\mathcal{Y}}:G\rightarrow GL\left(\mathcal{Y}\right)$ actions of $G$ such that

$$ $\Phi_{\textit{strict}}\left(\rho^{\mathcal{X}}\left(g\right)\mathbf{x}\right)= \rho^{\mathcal{Y}}\left(g\right)\Phi_{\textit{strict}}\left(\mathbf{x}\right),$ (1) $$

where
${GL}(\cdot)$ is a general linear group over the space $\cdot$.

Definition 2 (Relaxed Equivariance) []. A learning function $\Phi_{\textit{relaxed}}:\mathcal{X}\rightarrow\mathcal{Y}$ that sends elements from input space $\mathcal{X}$ to output space $\mathcal{Y}$ satisfies Relaxed Equivariance to a group $G$ if $\forall g,\mathbf{x}\in G\times\mathcal{X}$ there exists $\rho^{\mathcal{X}}:G\rightarrow GL\left(\mathcal{X}\right)$ and $\rho^{\mathcal{Y}}:G\rightarrow GL\left(\mathcal{Y}\right)$ actions of $G$ such that

$$ $\left\|\rho^{\mathcal{Y}}\left(g\right)\Phi_{\textit{relaxed}}\left(\mathbf{x} \right)-\Phi_{\textit{relaxed}}\left(\rho^{\mathcal{X}}\left(g\right)\mathbf{x }\right)\right\|<\epsilon,$ (2) $$

where $\epsilon$ is a controllable variable, with small $\epsilon$ exhibiting strong symmetry, while a lot larger $\epsilon$ exhibits greater flexibility and generalization ability, as building relaxed convolution filters can be learned from training data.
Note that, $\Phi_{\textit{relaxed}}$ is equivalent to $\Phi_{\textit{strict}}$ when $\epsilon=0$.
Meanwhile, $\Phi_{\textit{strict}}$ satisfies the relaxed equivariance condition.

Strictly Equivariant Neural Networks.
Learning equivariant features is an optimization process for a series of $\Phi_{\textit{strict}}$ function sets in the model.
Since the composition of equivariant functions is also equivariant, constructing a strict equivariant network is a composition of $\Phi_{\textit{strict}}$.
Simply put, if linear, nonlinear, pooling, aggregation, normalization, and other operators in a network are all equivariant, then their composite operators are also equivariant.
However, the challenge of strictly equivariant networks lies in designing trainable linear layers, such as equivariant convolutions.
Usually, there are two strategies for designing equivariant convolutions: weight sharing and weight typing, which are G-CNN [] and G-steerable CNN [], respectively.

Relaxed Equivariant Neural Networks.
The existing equivariant networks assume that the data is completely symmetric. This network approximates a strictly invariant or equivariant function under given group actions. For example, in G-CNN, the shared convolution filter achieves equivariant images at 0 0, $90$, $180$, and $270$ degrees under the strict constraint of the rotation group $\mathbf{C}_{4}$. However, real-world data is rarely symmetric. This seriously hinders the potential application of equivariant networks. To solve this problem, we can relax the strict constraints on weights under group actions. In [, ], relaxing weight constraints can significantly improve the performance and generalization ability of the model.

### 3.2 The Implementation of Relaxed Rotation-Equivariant GConv (R2GConv)

Relaxing strict group constraints is an effective way to achieve our Relaxed Rotation-Equivariant convolution filters. This paper proposes implementing a learnable parameter $\Delta$ to perturb the group operations.
Here, our proposed method is based on the fourth-order cyclic rotation group $\mathbf{C}_{4}=\{\mathbf{c}^{0},\mathbf{c}^{1},\mathbf{c}^{2},\mathbf{c}^{3}\}$,
where powers of $\mathbf{c}$ indicate performing rotation operation on the input $\mathbf{x}$ by $90$ degrees multiple times.
The affine transformation matrix $\mathcal{A}_{i}^{\mathbf{c}}$ on $\mathbf{C}_{4}$ can be defined as follows:

$$ $\mathcal{A}_{i}^{\mathbf{c}}=\left[\begin{array}[]{cc}\cos\left(\pi i/2\right) &-\sin\left(\pi i/2\right)\\ \sin\left(\pi i/2\right)&\cos\left(\pi i/2\right)\\ \end{array}\right],i\in\{0,1,2,3\}.$ (3) $$

Then, define $\Delta\in\mathbb{R}^{4\times 2\times 2}$ denotes our learnable perturbation factor, we have

$$ $\Delta=\{\Delta_{i}\}=\left\{\left[\begin{array}[]{cc}\Delta_{i1}&\Delta_{i2} \\ \Delta_{i3}&\Delta_{i4}\\ \end{array}\right]\right\}.$ (4) $$

Now, we define a Relaxed Rotation-Equivariant group $\mathbf{R}_{4}=\left\{\mathbf{r}^{0},\mathbf{r}^{1},\mathbf{r}^{2},\mathbf{r}^
{3}\right\}$ based on $\mathbf{C}_{4}$ and a function $\mathcal{T}:\mathbf{C}_{4}\rightarrow\mathbf{R}_{4}$.
We have a corresponding perturbed affine transformation matrix $\mathcal{A}_{i}^{\mathbf{r}}$ on $\mathbf{R}_{4}$,

$$ $\mathcal{A}_{i}^{\mathbf{r}}=\mathcal{T}(\mathcal{A}_{i}^{\mathbf{c}},\Delta_{ i})=\left[\begin{array}[]{cc}\cos\left(\pi i/2\right){\mathbf{+}}\Delta_{i1}&- \sin\left(\pi i/2\right){\mathbf{+}}\Delta_{i2}\\ \sin\left(\pi i/2\right){\mathbf{+}}\Delta_{i3}&\cos\left(\pi i/2\right){ \mathbf{+}}\Delta_{i4}\\ \end{array}\right],$ (5) $$

where we use simple addition to add noise to the affine transformation. Note that other operations are also available for $\mathcal{T}$, such as multiplication but are not limited to.
Meanwhile, for all $[a\ b]\in CoorSet(\mathbf{x})$,
we have the transformations $\mathbf{c}^{i}([a\ b])=\mathcal{A}_{i}^{\mathbf{c}}\cdot[a\ b]^{\top}$ and $\mathbf{r}^{i}([a\ b])=\mathcal{A}_{i}^{\mathbf{r}}\cdot[a\ b]^{\top}$ on $\mathbf{C}_{4}$ and on $\mathbf{R}_{4}$, respectively.
Note that $CoorSet(\mathbf{x})$ denotes the set of coordinates of $\mathbf{x}$.

Now we demostrate the construction of our proposed Relaxed Rotation-Equivariant convolution filters, which is the core of achieving catching symmetric-breaking feartures.

Given an initial $2$D convolution filter $\mathcal{K}_{init}$,
for all $[u\ v]\in CoorSet(\mathcal{K}_{init})$, and $\mathbf{r}^{i}\in\mathbf{R}_{4}$ where $i\in\{0,1,2,3\}$,
the transformed coordinates of Relaxed Rotation-Equivariant convolution filters are given as follows:

$$ $\begin{split}\begin{bmatrix}u^{\prime}\\ v^{\prime}\end{bmatrix}={\mathbf{r}^{i}([u\ v])}&=\left[\begin{array}[]{cc} \cos\left(\pi i/2\right){\mathbf{+}}\Delta_{i1}&-\sin\left(\pi i/2\right){ \mathbf{+}}\Delta_{i2}\\ \sin\left(\pi i/2\right){\mathbf{+}}\Delta_{i3}&\cos\left(\pi i/2\right){ \mathbf{+}}\Delta_{i4}\\ \end{array}\right]\begin{bmatrix}u\\ v\end{bmatrix}.\end{split}$ (6) $$

We rounded the non-integer coordinates $[u^{\prime}\ v^{\prime}]$ to the nearest integer in Equation 6.
Further, if $[u^{\prime}\ v^{\prime}]\in CoorSet(\mathcal{K}_{rel})$, we set $\mathcal{K}_{rel}[u^{\prime}\ v^{\prime}]:=\mathcal{K}_{init}[u\ v]$. All the coordinates of the missing values within $\mathcal{K}_{rel}$ will be performed through Bilinear interpolation.

For the detailed construction of $\mathcal{K}_{rel}$, we provide a pseudocode as shown in Algorithm 1.

Figure: Algorithm 1 Build Relaxed Rotation-Equivariant convolution filters based on $\mathbf{C}_{4}$.

So far, we have achieved our Relaxed Rotation-Equivariant convolution filter by relaxing the strict constraints of group operations on $\mathbf{C}_{4}$ through our learnable perturbation factor $\Delta$.

Based on it, our following content will introduce Relaxed Rotation-Equivariant GConv (R2GConv), which is divided into two convolutions: Relaxed Rotation-Equivariant Pointwise GConv (R2PGConv) and Relaxed Rotation-Equivariant Depthwise GConv (R2DGConv). The reason for dividing R2GConv into two operators, pointwise and depthwise [], is to reduce the enormous number of parameters and computational overhead brought by GConv. For convenience, we define the abovementioned algorithm as an operator $\Phi$ for generating Relaxed Rotation-Equivariant convolution filters.

Relaxed Rotation-Equivariant Pointwise GConv (R2PGConv).
Given that the input image feature map $\mathbf{f}_{in}$ has been projected into the space $\mathbf{R}_{4}$ with a size of $c_{in}\times 4\times h\times w$.
Given an initial convolution filter $\mathcal{K}_{init}^{p}$ with a size of $c_{out}\times c_{in}\times 4\times 1\times 1$, we can obtain our Relaxed Rotation-Equivariant pointwise convolution filter $\mathcal{K}_{rel}^{p}=\Phi\left(\mathcal{K}_{init}^{p}\right)$ with a size of $c_{out}\times 4\times c_{in}\times 4\times 1\times 1$. Then, our R2PGConv can be defined as

$$ $\mathbf{f}_{out}\left({c_{out},4,h,w}\right)=\sum_{c}^{c_{in}}\sum_{g}^{4} \mathcal{K}_{rel}^{p}\left(c_{out},4,c,g,1,1\right)\star\mathbf{f}_{in}\left(c ,g,h,w\right),$ (7) $$

where the size of $\mathbf{f}_{out}$ is $c_{out}\times 4\times h\times w$, and the operator $\star$ denotes group convolution operation. It is particularly noteworthy that during the convolution operation, $\mathcal{K}_{rel}^{p}$ will be reshaped to $4c_{out}\times 4c_{in}\times 1\times 1$, $\mathbf{f}_{in}$ will be reshaped to $4c_{in}\times h\times w$ to meet the input format of the convolution. The output also will be reshaped to $c_{out}\times 4\times h\times w$. The construction of the convolution filter for R2PGConv is detailed in Figure 2.

Figure: Figure 2: (a): The construction of our Relaxed Rotation-Equivariant convolution filters using the pointwise and depthwise convolution operators. (b): The composition of our R2GConv.
Refer to caption: x2.png

Relaxed Rotation-Equivariant Depthwise GConv (R2DGConv).
To reduce the huge computational cost of GConv, our R2DGConv adopted the grouped convolution operator.
Similar to above, given an initial convolution filter $\mathcal{K}_{init}^{d}$ with a size of $c_{out}\times 1\times 1\times k\times k$, we can obtain our Relaxed Rotation-Equivariant depthwise convolution filter $\mathcal{K}_{rel}^{d}=\Phi\left(\mathcal{K}_{init}^{d}\right)$ with a size of $c_{out}\times 4\times 1\times 1\times k\times k$. Similarly, our R2DGConv can be defined as

$$ $\widetilde{\mathbf{f}}_{out}\left({c_{out},4,h^{\prime},w^{\prime}}\right)= \mathcal{K}_{rel}^{d}\left(c_{out},4,1,1,k,k\right)\star_{gd}\mathbf{f}_{out} \left(c_{out},4,h,w\right),$ (8) $$

where the size of $\widetilde{\mathbf{f}}_{out}$ is $c_{out}\times 4\times h^{\prime}\times w^{\prime}$, the operator $\star_{gd}$ denotes the grouped convolution, and its number of groups is $4c_{out}$. Also, during the convolution process, $\mathcal{K}_{rel}^{d}$ will be reshaped to $4c_{out}\times 1\times k\times k$, $\mathbf{f}_{out}$ will be reshaped to $4c_{out}\times h\times w$.

At this point, R2GConv can be defined as follows:

$$ $\widetilde{\mathbf{f}}_{out}=\mathcal{K}_{rel}^{d}\star_{gd}\left(\mathcal{K}_ {rel}^{p}\star\mathbf{f}_{in}\right).$ (9) $$

Here, the pointwise operation is mainly used for connecting channels, while the depthwise operation reduces the high computational cost. The motivation behind using the pointwise operation first and then the depthwise operation is that the pointwise operation can reduce the number of input channels, thereby reducing the number of the groups of the grouped convolution in the depthwise operation, which ultimately reduces the total parameters of our R2GConv.

### 3.3 The Relaxed Rotation-Equivariant Network (R2Net)

Figure: Figure 3: The architecture of our R2Net (Backbone).
Refer to caption: x3.png

Figure: Figure 4: The architecture of our (a) R2Net Block, (b) Bottleneck, (c) R2GConv3x3, and (d) Transfer Block. The format of GCBA3x3 means GConv3x3$\rightarrow$ BatchNorm $\rightarrow$ Activate.
Refer to caption: x4.png

Based on our R2GConv, we propose a Relaxed Rotation-Equivariant Network (R2Net).
Our R2Net (BackBone) consists of four stages: a downsampled R2GCBA3x3 and an R2Net Block.
Since the input of a regular CNN is a 2D feature map in the plane $\mathbb{Z}^{2}$ space, we typically project the input to $\mathbf{R}_{4}$ through our proposed Relaxed Rotation-Equivariant Lifting (R2Lifting) in the first layer. Given a 2D feature map $\mathbf{f}_{in}$ with a size of $c_{in}\times h\times w$ and an initial convolution filter $\mathcal{K}_{init}^{l}$ with a size of $c_{out}\times c_{in}\times k\times k$, we can obtain our Relaxed Rotation-Equivariant lifting convolution filter $\mathcal{K}_{rel}^{l}=\Phi\left(\mathcal{K}_{init}^{l}\right)$ with a size of $c_{out}\times 4\times c_{in}\times k\times k$. Hence, R2Lifting can be defined as

$$ $\mathbf{f}_{out}\left({c_{out},4,h^{\prime},w^{\prime}}\right)=\sum_{c}^{c_{in }}\mathcal{K}_{rel}^{l}\left(c_{out},4,c,k,k\right)\star\mathbf{f}_{in}\left(c ,h,w\right),$ (10) $$

where the size of $\mathbf{f}_{out}$ is $c_{out}\times 4\times h^{\prime}\times w^{\prime}$, in which $h^{\prime}$ and $w^{\prime}$ denote the output height and width, respectively. Again, during the convolution process, $\mathcal{K}_{init}^{l}$ will be reshaped to $4c_{out}\times c_{in}\times k\times k$, and $\mathbf{f}_{in}$ remains unchanged. Then, input the projected feature map into the $4$-layer stage, perform $2\times$ downsampling in sequence, and increase the number of channels by $2\times$. Specifically, the last stage preserves the number of channels unchanged. The obtained output will go through the Transfer Block and then be sent to a universal classification header.
The architecture of our R2Net is shown in Figure 3, and the Transfer Block is shown in Figure 4 (d).
We provided three models of different sizes in this paper: R2Net-n, R2Net-s, and R2Net-m.
More details can be found in Appendix A.1.

Due to the inevitable high computational overhead caused by GConv, we borrow the idea of dividing channels in Res2Net [] to reduce the number of parameters and FLOPs.
The R2Net Block is designed with the residual connection [] architecture to improve the convergence ability.
The detailed module architecture of R2Net Block is shown in Figure 4 (a).

### 3.4 The Redesigned Symmetry-Breaking Object Detector (SBDet)

Figure: Figure 5: The architecture of SBDet-m.
Refer to caption: x5.png

Based on our proposed R2Net, we propose a Symmetry-Breaking Object Detector (SBDet). SBDet adopts the FPN+PAN neck architecture, taking the stage outputs $2\sim 4$ in R2Net as inputs. Specifically, the input of stage $4$ will go through a GSPPF for spatial max pooling under the group $\mathbf{C}_{4}$. The obtained inputs are processed through the neck network to obtain multi-scale features, which are then fed into the Transfer Block. Finally, the outputs are fed into a universal detection head of YOLOv8, including three detection heads for detecting small, medium, and large-scale objects.
Note that the R2GUp adopts the same architecture as R2GConv, except that the transposed convolution operator is used during the depthwise convolution.
The detailed architecture is shown in Figure 5.

## 4 Experiments

In this section, we conduct extensive experiments to study and demonstrate the performance of the proposed model.
In Sec. 4.1, we evaluate our proposed method on the PASCAL VOC and MS COCO 2017 datasets, showing the state-of-the-art performance on the object detection task.
We also conduct natural image classification experiments on CIFAR-10/100 dataset, as described in Sec. 4.2.
To provide further insight into the effectiveness of our approach, we also present visualization results in Sec. 4.3.
The experimental results show that our model has a greater parameter efficiency and accuracy capacity than the existing strict ENNs.
We adopt our Relaxed Rotation-Equivariant group $\mathbf{R}_{4}$ to construct all our models in this section, and a declaration will be made in case of special settings.
All the experiments are done on dual GeForce RTX 4090 GPUs.

### 4.1 Object Detection on the PASCAL VOC and MS COCO Datasets

PASCAL Visual Object Classes (VOC) 2007 comprises 5,000 training images with over 12,000 annotated objects.
The PASCAL VOC 2012 expands this with 11,000 training images and 27,000 annotated objects.
Both datasets annotate 20 everyday object classes, including person, cat, bicycle, etc.
We conduct experiments on the combined PASCAL VOC 07$\&$12.
Furthermore, experiments were extended to the Microsoft Common Objects in Context (MS COCO) 2017 dataset, comprised of 164,000 images annotated with 897,000 objects across 80 classes.
We aim to validate the capability of our proposed SBDet in standard object detection tasks using mean Average Precision (mAP). Specifically, we leverage $\textbf{mAP}_{50}$ at an intersection over union (IoU) threshold of 0.5, and $\textbf{mAP}_{50:90}$ across IoU thresholds ranging from 0.5 to 0.95 as key evaluation metrics.

Figure: Table 1: Ablation studies on R.R.E. and S.R.E. in YOLOv8-n and SBDet-n on PASCAL VOC07$\&$12.

Figure: Table 2: Performance of SBDet-n on PASCAL VOC 07$\&$12 for different variance $\sigma$.

We conduct ablation analyses on Relaxed Rotation Equivariance (R.R.E.) and Strict Rotation Equivariance (S.R.E.) in YOLOv8-n and our proposed SBDet-n.
As shown in Table 4.1, when both R.R.E. and S.R.E. are disabled, the baseline $\textbf{mAP}_{50}$ scores of YOLOv8-n and SBDet-n are $78.6$ and $77.9$ respectively. When enabling R.R.E. (or S.R.E.), the $\textbf{mAP}_{50}$ scores increase by $5\%$ and $6.2\%$ (or $4.1\%$ and $5.1\%$) respectively, indicating that R.R.E. and S.R.E. improve the accuracy of object detection. However, the $\textbf{mAP}_{50}$ improvement of R.R.E. is greater than that of S.R.E. both in SBDet-n and YOLOv8-n, indicating the effectiveness of our R.R.E. The same situation also occurs on the $\textbf{mAP}_{50:95}$ improvement. To investigate further the effect of different initial parameters $\sigma$ on the performance of SBDet-n, we perform experiments with various values of $\sigma$ ranging from $0.1$ to $0.8$.
As shown in Table 4.1, when $\sigma$ is set to 0.1, 0.2, or 0.4, mAP fluctuates little, but when $\sigma$ is set to 0.6 or 0.8, mAP decreases significantly.
Refer to Appendix A.2 for model convergence details across various $\sigma$.
Note that $\sigma$ is set to $0.1$ in the following.

Table 4.1 presents the PASCAL VOC 07$\&$12 test results.
Notably, our SBDet-m ($22.6$M) stands out with its remarkable performance, boasting the highest $87.3\%$ $\textbf{mAP}_{50}$, and $70.6\%$ $\textbf{mAP}_{50:95}$ among the listed models. Notably our SBDet-m achieves an increase in $\textbf{mAP}_{50}$ (and $\textbf{mAP}_{50:95}$) by $0.4\%$ (and $1.1\%$) compared to YOLOv8-x, also having only $33.1\%$ parameters of YOLOv8-x ($68.2$M). In addition, our SBDet-n has the least parameters but with good results.

We also conduct experiments on a larger dataset, MS COCO 2017, as shown in Table 4.1.
Our SBDet-n ($2.8$M) achieved a surprising improvement in $\textbf{mAP}_{50:95}$ (and $\textbf{mAP}_{50}$) by $6.2\%$ (and $6.4\%$) compared to YOLOv6-n, despite having only $59.6\%$ parameters of YOLOv6-n ($4.7$M). Furthermore, our SBDet-s achieved slightly lower (or equal) $\textbf{mAP}_{50:95}$ and $\textbf{mAP}_{50}$ compared to YOLOV8-m (or YOLOv6-m), but only with about $33.3\%$ of their parameters. Similarly, our SBDet-m achieved an approximate $\textbf{mAP}_{50}$ and $\textbf{mAP}_{50:95}$ compared to YOLOv8-l, but with about a half parameters of YOLOv8-l. It can be seen that our SBDet has fewer parameters and higher accuracy.

SBDet excels in the trade-off between efficiency and accuracy. Further details on convergence analyses and loss curve experiments are provided in Appendix A.3.

Figure: Table 3: PASCAL VOC 07$\&$12 test results.

Figure: Table 4: MS COCO 2017 validation results.

### 4.2 Image Classification on the CIFAR-10/100 and Rotated MNIST Datasets

In this section, we evaluate the performance of our proposed R2Net on image classification tasks using the CIFAR-10/100 and Rotated MNIST datasets, where we randomly rotate all images in the MNIST training set by 0, 90, 180, and 270 degrees, leaving the test set unaffected.
As seen from Table 4.2, our R2Net-n achieves competitive results compared to other models while having much smaller parameters (0.8M).
Our R2Net-m achieved a top-1 test error rate of only $3.5\%$ on CIFAR-10 and $17.3\%$ on CIFAR-100, slightly lower than ResNeXt-29, but with only 6M parameters, which is $10\%$ parameters of ResNeXt-29. Furthermore, our R2Net-x outperforms all other models with a top-1 test error rate of only $3.33\%$ on CIFAR-10 and $16.1\%$ on CIFAR-100.
Additionally, we also evaluate the performance of our R2Net on the Rotated MNIST dataset.
As shown in Table 4.2, Our R2Net-n achieves a top-1 test error rate of $27.75\%$, outperforming YOLOv8-n-cls in recognizing rotated images.
Overall, these results demonstrate the superior performance of our proposed R2Net in image classification tasks on both standard and rotated datasets.

Figure: Table 5: Top-1 test error (%) on CIFAR-10/100 dataset.

Figure: Table 6: Top-1 test error (%) on MNIST (M.) and Rotated MNIST (R.M.) dataset.

### 4.3 Relaxed, Strict Rotation-Equivariance and Non-Rotation-Equivariance Visualization

The visualization of our R.R.E., S.R.E., and Non-Rotation-Equivariance (N.R.E.) is shown in Figure 6. We rotate the original image (a) to obtain rotated images (b) and input them into our SBDet (R.R.E.), SBDet (S.R.E.), and YOLOv8 (N.R.E.) to obtain the feature maps (c), (d), and (e), respectively. Observing the white circles in (c), we notice slight differences. Nevertheless, the overall feature maps are Rotation-Equivariant, showcasing our R.R.E. Observing the red circles in (d), we find that the feature maps are strictly equivariant, presenting S.R.E. Lastly, observing (e), we find almost N.R.E.
A more detailed visualization of R.R.E. in our SBDet-n can be found in Appendix A.6.

Figure: Figure 6: Visualization of feature maps in SBDet-n (R.R.E. and S.R.E.) and YOLOv8-n (N.R.E.).
Refer to caption: x6.png

## 5 Conclusion

In this work, we propose Relaxed Rotation-Equivariant Group Convolution (R2GConv), a novel approach that tackles symmetry-breaking situations in rotation-equivariance to better align with real-world scenarios.
R2GConv introduces learnable parameters to transform a strict rotation-equivariant group $\mathbf{C}_{4}$ into a relaxed rotation-equivariant group $\mathbf{R}_{4}$ by perturbing group operations.
Furthermore, we propose an efficient Relaxed Rotation-Equivariant Network (R2Net) as the backbone and a redesigned $2$D object detector called Symmetry-Breaking Object Detector (SBDet).
Through experiments on the object detection and classification tasks, we demonstrate that our proposed SBDet and R2Net achieve state-of-the-art performance compared to models without symmetry bias or with strict equivariant constraints.
However, a limitation is the relatively slower training speed, so integrating CUDA-accelerated operations is one of the future directions.
Additionally, R2GConv or R2Net can be applied to more complex visual tasks and scenes, leveraging the advantages of Relaxed Rotation-Equivariance.

## Appendix A Appendix

### A.1 Model architecture of SBDet

In this section, we provide a detailed construction of our SBDets of different sizes, as shown in Table 7.

**Table 7: The model architecture of our SBDet-n, SBDet-s, and SBDet-m. Among them, the NO. column shows the number of layers; The From column indicates where the input of this layer module comes from. For example, -1 represents the upper layer, -1, 6 represents the upper layer, and layer 6. Note that in the Output Size column, $\times 4$ denotes the group dimension, not the number of channels. O.C. denotes Output channels in the table below.**
| No. | From | Module | Output Size | SBDet-n | SBDet-s | SBDet-m |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| O.C. | Params | O.C. | Params | O.C. | Params |  |  |  |  |
| 0 | -1 | R2Lifting | $4\times 320\times 320$ | 16 | 480 | 32 | 944 | 48 | 1408 |
| 1 | -1 | R2GCBA3x3 | $4\times 160\times 160$ | 32 | 2416 | 64 | 8912 | 96 | 19504 |
| 2 | -1 | R2Net Block | $4\times 160\times 160$ | 32 | 9640 | 64 | 37680 | 96 | 131152 |
| 3 | -1 | R2GCBA3x3 | $4\times 80\times 80$ | 64 | 8912 | 128 | 34192 | 192 | 75856 |
| 4 | -1 | R2Net Block | $4\times 80\times 80$ | 64 | 58784 | 128 | 232192 | 192 | 409664 |
| 5 | -1 | R2GCBA3x3 | $4\times 40\times 40$ | 128 | 34192 | 256 | 133904 | 384 | 299152 |
| 6 | -1 | R2Net Block | $4\times 40\times 40$ | 128 | 232192 | 256 | 923072 | 384 | 1630208 |
| 7 | -1 | R2GCBA3x3 | $4\times 20\times 20$ | 128 | 66960 | 256 | 264976 | 384 | 594064 |
| 8 | -1 | R2Net Block | $4\times 20\times 20$ | 128 | 149056 | 256 | 592992 | 384 | 2072704 |
| 9 | -1 | GSPPF | $4\times 20\times 20$ | 128 | 65792 | 256 | 262656 | 384 | 590592 |
| 10 | -1 | R2GUp | $4\times 40\times 40$ | 64 | 33040 | 128 | 131600 | 192 | 295696 |
| 11 | -1, 6 | GConcat | $4\times 40\times 40$ | 192 | 0 | 384 | 0 | 576 | 0 |
| 12 | -1 | R2Net Block | $4\times 40\times 40$ | 128 | 181824 | 256 | 724064 | 384 | 2515072 |
| 13 | -1 | R2GUp | $4\times 80\times 80$ | 64 | 33040 | 128 | 131600 | 192 | 295696 |
| 14 | -1, 4 | GConcat | $4\times 80\times 80$ | 128 | 0 | 256 | 0 | 384 | 0 |
| 15 | -1 | R2Net Block | $4\times 80\times 80$ | 64 | 54064 | 128 | 214592 | 192 | 741472 |
| 16 | -1 | R2GCBA3x3 | $4\times 40\times 40$ | 64 | 17104 | 128 | 66960 | 192 | 149584 |
| 17 | -1, 12 | GConcat | $4\times 40\times 40$ | 192 | 0 | 384 | 0 | 576 | 0 |
| 18 | -1 | R2Net Block | $4\times 40\times 40$ | 128 | 181824 | 256 | 724064 | 384 | 2515072 |
| 19 | -1 | R2GCBA3x3 | $4\times 20\times 20$ | 128 | 66960 | 256 | 264976 | 384 | 594064 |
| 20 | -1, 9 | GConcat | $4\times 20\times 20$ | 256 | 0 | 512 | 0 | 768 | 0 |
| 21 | -1 | R2Net Block | $4\times 20\times 20$ | 256 | 592992 | 512 | 2365600 | 576 | 5320864 |
| 22 | 15 | Transfer Block | $80\times 80$ | 64 | 4288 | 128 | 16752 | 192 | 37408 |
| 23 | 18 | Transfer Block | $40\times 40$ | 128 | 16752 | 256 | 66256 | 384 | 148528 |
| 24 | 21 | Transfer Block | $20\times 20$ | 256 | 66256 | 512 | 263568 | 576 | 333376 |
| 25 | 22, 23, 24 | Detector Head | - | - | 897664 | - | 2147008 | - | 3822016 |
| Total Params |  | 2.8M |  | 9.6M |  | 22.6M |  |  |  |
| FLOPs |  | 4.0G |  | 9.0G |  | 17.3G |  |  |  |

### A.2 Convergence Analysis on SBDet models with different σ 𝜎 \sigma italic_σ

In this section, we further conduct a convergence analysis on the learnable perturbation factor by examining the impact of varying the variance $\sigma$ of its initial Gaussian distribution in our SBDet-n model on the PASCAL VOC dataset.
As illustrated in Figure 7,
it is clear that when $\sigma$ values are set to $0.1$, $0.2$, and $0.4$, the convergence behavior of the model remains consistent across these values, ultimately converging to similar accuracy in $\textbf{mAP}_{50}$ (and $\textbf{mAP}_{50:95}$).
The highest accuracy is observed with the variance $\sigma=0.1$.
However, for $\sigma$ values set to 0.6 and 0.8, there is a lot of decrease in model accuracy, particularly with $\sigma=0.8$ exhibiting the lowest accuracy.
We speculate that for larger initial $\sigma$, there is too much perturbation to the group operation, which leads to a significant degree of Symmetry-Breaking in the model, making it difficult for the model to learn the appropriate degree of Symmetry-Breaking from the training dataset, resulting in lower accuracy.

Figure: Figure 7: Comparison of SBDet-n with different $\sigma$. All models were trained for $200$ epochs on dual 4090 GPUs.
Refer to caption: x7.png

### A.3 Supplymental Convergence Analysis on SBDet models

In this section, we analyze the accuracy and loss curves of the SBDet and YOLOv8 models during the training process on the PASCAL VOC dataset, as detailed in Figure 8.
In part (a), it is evident that our SBDet model exhibits not only a more stable training process but also surpasses the accuracy of outstanding YOLOv8 models.
Moreover, our model achieves faster convergence, significantly reducing the total number of training epochs needed to reach the target accuracy, which it attains within approximately $90$ to $100$ epochs.
The fast convergence could be related to the fact that our R2GCConv can extract rich and relaxed equivariant features, hence enabling earlier learning of these potential features.
Regarding the loss function, both YOLOv8 and SBDet exhibit a steady decline across all three loss components, as illustrated in parts (b), (c), and (d) of the figure.

Figure: Figure 8: Comparison of YOLOv8 and our SBDet models. All models were trained on the PASCAL VOC dataset for $200$ epochs with dual 4090 GPUs.
Refer to caption: x8.png

### A.4 Supplymental Experiments on PASCAL VOC 07 & \& & 12 dataset

As shown in Table 8, AP for different $20$ classes of the PASCAL VOC 07$\&$12 are reported.
SBDet demonstrates a balanced and consistently high performance across various categories.
Even in categories where SBDet does not achieve the highest AP, it is still competitive compared to models of comparable size.

**Table 8: PASCAL VOC 07$\&$12 test results. The highest APs for each class are indicated in bold.**
|  |  | AP of 20 classes |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Method | mAP | Aero | Bike | Bird | Boat | Bottle | Bus | Car | Cat | Chair | Cow | Table | Dog | Horse | Mbike | Person | Plant | Sheep | Sofa | Train | TV |
| Faster-RCNN | 76.4 | 79.8 | 80.7 | 76.2 | 68.3 | 55.9 | 85.1 | 85.3 | 89.8 | 56.7 | 87.8 | 69.4 | 88.3 | 88.9 | 80.9 | 78.4 | 41.7 | 78.6 | 79<br>.8 | 85.3 | 72.0 |
| R-FCN | 80.5 | 79.9 | 87.2 | 81.5 | 72.0 | 69.8 | 86.8 | 88.5 | 89.8 | 67.0 | 88.1 | 74.5 | 89.8 | 90.6 | 79.9 | 81.2 | 53.7 | 81.8 | 81.5 | 85.9 | 79.9 |
| SSD300 | 77.5 | 79.5 | 83.9 | 76.0 | 69.6 | 50.5 | 87.0 | 85.7 | 88.1 | 60.3 | 81.5 | 77.0 | 86.1 | 87.5 | 83.9 | 79.4 | 52.3 | 77.9 | 79.5 | 87.6 | 76.8 |
| SSD512 | 79.5 | 84.8 | 85.1 | 81.5 | 73.0 | 57.8 | 87.8 | 88.3 | 87.4 | 63.5 | 85.4 | 73.2 | 86.2 | 86.7 | 83.9 | 82.5 | 55.6 | 81.7 | 79.0 | 86.6 | 80.0 |
| DSSD321 | 78.6 | 81.9 | 84.9 | 80.5 | 68.4 | 53.9 | 85.6 | 86.2 | 88.9 | 61.1 | 83.5 | 78.7 | 86.7 | 88.7 | 86.7 | 79.7 | 51.7 | 78.0 | 80.9 | 87.2 | 79.4 |
| DSSD513 | 81.5 | 86.6 | 86.2 | 82.6 | 74.9 | 62.5 | 89.0 | 88.7 | 88.8 | 65.2 | 87.0 | 78.7 | 88.2 | 89.0 | 87.5 | 83.7 | 51.1 | 86.3 | 81.6 | 85.7 | 83.7 |
| YOLOv8-n | 78.6 | 86.5 | 88.0 | 75.1 | 69.9 | 65.3 | 85.3 | 90.6 | 85.3 | 61.0 | 81.0 | 75.3 | 81.1 | 89.2 | 86.0 | 87.0 | 50.9 | 77.2 | 73.4 | 88.2 | 76.2 |
| YOLOv8-s | 81.6 | 90.7 | 89.1 | 80.2 | 72.8 | 69.4 | 89.0 | 92.3 | 88.1 | 63.5 | 85.3 | 75.6 | 84.9 | 92.1 | 89.9 | 88.4 | 55.1 | 80.9 | 77.0 | 89.4 | 79.2 |
| YOLOv8-m | 83.7 | 92.2 | 92.7 | 83.2 | 74.7 | 73.6 | 89.1 | 92.7 | 91.2 | 67.4 | 87.0 | 78.5 | 87.3 | 92.7 | 90.4 | 89.2 | 59.5 | 80.6 | 80.7 | 89.9 | 81.7 |
| YOLOv8-l | 86.4 | 95.0 | 93.7 | 84.4 | 78.2 | 78.0 | 92.6 | 94.2 | 92.7 | 71.0 | 89.6 | 81.6 | 89.3 | 94.3 | 92.2 | 90.8 | 61.9 | 87.0 | 83.4 | 90.8 | 86.7 |
| YOLOv8-x | 86.9 | 93.5 | 93.9 | 86.1 | 79.1 | 77.5 | 92.6 | 94.3 | 92.2 | 72.3 | 89.8 | 83.2 | 90.9 | 94.7 | 92.5 | 90.8 | 64.5 | 87.7 | 84.3 | 91.7 | 86.3 |
| SBDet-n | 84.1 | 92.4 | 90.7 | 85.1 | 78.0 | 74.1 | 89.4 | 92.7 | 90.1 | 67.9 | 86.2 | 79.8 | 88.0 | 93.0 | 89.4 | 90.2 | 58.1 | 84.7 | 78.9 | 89.3 | 83.3 |
| SBDet-s | 86.0 | 94.0 | 93.0 | 85.6 | 79.1 | 76.3 | 91.0 | 94.0 | 92.8 | 69.8 | 89.1 | 80.6 | 90.6 | 93.1 | 91.7 | 91.7 | 64.1 | 86.6 | 79.8 | 91.7 | 84.3 |
| SBDet-m | 87.3 | 94.6 | 94.5 | 89.3 | 79.9 | 78.9 | 93.1 | 94.3 | 94.8 | 71.6 | 90.1 | 81.2 | 92.8 | 93.3 | 92.5 | 92.0 | 64.8 | 89.2 | 82.0 | 91.6 | 85.5 |

### A.5 Analysis on YOLOv8-n-cls and R2Net-n on the Rotated MINIST (R.M.) dataset

This section compares the training accuracy of YOLOv8-n-cls and our R2Net-n on the R.M. dataset.
We manipulate the training set by randomly rotating $60,000$ images by 0 0, $90$, $180$, and $270$ degrees while maintaining $10,000$ images unaltered in the test set to evaluate the performance of a model under rotation.
As depicted in Figure 9, both R2Net-n and YOLOv8-n-cls display fluctuations during training.
However, R2Net-n exhibits milder fluctuations compared to the more pronounced oscillations observed in YOLOv8-n-cls.
This contrast highlights the superior rotational anti-interference capability of R2Net-n, which is primarily attributed to its novel Relaxed Rotation Equivariance (R.R.E.) property.

Figure: Figure 9: Comparison of YOLOv8-n and our R2Net-n in training accuracy on the R.M. dataset. Both models were trained for 50 epochs on dual 4090 GPUs with the resized input size $224\times 224$.
Refer to caption: x9.png

### A.6 Visualization Analysis on Relaxed Rotation-Equivariance

In this subsection, we present a visualization of feature maps from our SBDet-n, as illustrated in Figure 10.
We rotate the initial image (a) by 90, 180, and 270 degrees to generate images (b), (c), and (d) as inputs.
It can be found that the output feature maps in (e), (f), (g), and (h), corresponding to each channel, exhibit consistency with minor variations, which demonstrates the Relaxed Rotation-Equivariance property of our network.

Figure: Figure 10: The SBDet-n feature map visualization of the original image rotated at (a) 0, (b) 90, (c) 180, and (d) 270 degrees, as depicted in (e), (f), (g), and (h), corresponds to its $32$ channels.
Refer to caption: x10.png

### A.7 Heatmap Visualization

In this section, we present the visualization of LayerCAM [] heatmaps derived from YOLOv8-n, YOLOv7, YOLOv5, and our SBDet-n, as depicted in Figure 11.
These heatmaps enable us to locate the regions of interest where the network concentrates its attention.
It can be seen that YOLOv7 and our SBDet-n achieved better feature focusing.
Notably, SBDet-n shows a comprehensive focusing range on certain objects, such as dogs and zebras.
In contrast, YOLOv8 and YOLOv5 fail to exhibit such targeted feature focus on these particular objects.

Figure: Figure 11: LayerCAM Heatmaps Visualizations: Neck Networks across YOLO Models (v8-n, v7, v5) and our Novel SBDet-n Architecture. Examples are from MS COCO Dataset
Refer to caption: x11.png

### A.8 Parameter Analysis of R2GConv and GConv based on the Rotation group 𝐂 4 subscript 𝐂 4 \mathbf{C}_{4} bold_C start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT

Assuming the input channels, output channels, and kernel size of both R2GConv and GConv are $c_{in}$, $c_{out}$, and $k$, respectively. The parameters of our R2GConv can be calculated as follows:

$$ $\begin{split}c_{in}\times c_{out}\times 4\times 1\times 1\left(\textbf{R2 PGConv}\right)+c_{out}\times 1\times 1\times k\times k\left(\textbf{R2DGConv} \right)+4\times 2\times 2\left(\Delta\right)\\ \approx 4\times c_{in}\times c_{out}+k^{2}\times c_{out}\end{split}$ (11) $$

where $\cdot$ in $\left(\cdot\right)$ denotes the source of parameters.

The parameters of GConv can be calculated as follows:

$$ $c_{in}\times c_{out}\times 4\times k\times k=4\times c_{in}\times c_{out} \times k^{2}.$ (12) $$

Therefore, the parameter of our R2GConv is only

$$ $\frac{4\times c_{in}\times c_{out}+k^{2}\times c_{out}}{4\times c_{in}\times c _{out}\times k^{2}}=\frac{1}{k^{2}}+\frac{1}{4\times c_{in}}$ (13) $$

of GConv.

### A.9 Limitation on Training Speed and Memory

Although SBDet has significant advantages in accuracy and parameter efficiency, the relatively slow training speed during R2GConv is primarily due to the involved group transformation operations, PointWise, and Depthwise operators, lacking optimization in CUDA. Additionally, the R2Net Block in the model, which incorporates residual concatenation, consumes a considerable amount of memory usage during training. Moving forward, specific CUDA operators can be developed to address the training speed issue and explore better construction methods for the R2Net Block.

### A.10 Threotical Analysis

Since existing methods, including strictly rotation equivariant models, cannot perfectly tackle Symmetry-Breaking scenarios in object detection tasks, i.e., they can not learn an approximately equivariant function.
We assume from [] that
the ground truth equivariant function $\Phi_{gt}$ is approximately equivariant. A model class with a similar degree of approximate equivariance would better approximate $\Phi_{\textit{relaxed}}$ than $\Phi_{\textit{strict}}$ a strictly equivariant class or a class without bias towards symmetry.

Firstly, the Equivariance Error (EE) quantifies how much the ground truth equivariant function $\Phi_{gt}$ is approximately equivariant.
Let $\|\cdot\|$ denote the induced norm.

###### Definition A.1 (Equivariance Error [ 47 ] )

Let $\Phi_{gt}\colon\mathcal{X}\to\mathcal{Y}$ be a function and $G$ be a group. Assume that $G$ acts on $\mathcal{X}$ and $\mathcal{Y}$ via representation $\rho^{\mathcal{X}}$ and $\rho^{\mathcal{Y}}$. Then the Equivariance Error of $\Phi_{gt}$ is

$$ $\|\Phi_{gt}\|_{EE}=\sup_{x,g}\|\rho^{\mathcal{Y}}(g)\Phi_{gt}(x)-\Phi_{gt}( \rho^{\mathcal{X}}(g)(x))\|.$ $$

Hence, $\Phi_{gt}$ is $\epsilon$-approximately equivariant if and only if $\|\Phi_{gt}\|_{\mathrm{EE}}<\epsilon.$

###### Proposition A.2

Let $\Phi_{gt}$ be $\epsilon$-approximately equivariant and Lipschitz, with constant $k$. Then,

$$ $\displaystyle\|\rho^{\mathcal{Y}}(g)\cdot\Phi_{gt}(x)-\Phi_{gt}(x)\|\leq k\| \rho^{\mathcal{X}}(g)\cdot x-x\|+\epsilon,\forall g,x\in G\times\mathcal{X}.$ $$

Proof.
If $\Phi_{gt}$ is Lipschitz with constant $k$, we have

$$ $\displaystyle\|\Phi_{gt}(\rho^{\mathcal{X}}(g)\cdot x)-\Phi_{gt}(x)\|\leq k\| \rho^{\mathcal{X}}(g)\cdot x-x\|,\forall g,x\in G\times\mathcal{X}.$ $$

From relaxed equivariance of $g\cdot$, and triangle inequality, we find

$$ $\displaystyle\|\rho^{\mathcal{Y}}(g)\cdot\Phi_{gt}(x)-\Phi_{gt}(x)\|$ $\displaystyle\leq\|\rho^{\mathcal{Y}}(g)\cdot\Phi_{gt}(x)-\Phi_{gt}(\rho^{ \mathcal{X}}(g)(x))\|$ $\displaystyle\quad+\|\Phi_{gt}(\rho^{\mathcal{X}}(g)(x))-\Phi_{gt}(x)\|$ $\displaystyle\leq k\|\rho^{\mathcal{X}}(g)\cdot x-x\|+\epsilon,\forall g,x\in G \times\mathcal{X}.$ $$

$\hfill\square$

It is worth noting that both terms, $k\|\rho^{\mathcal{X}}(g)\cdot x-x\|$ and $\epsilon$, collectively determine the overall upper limit of the equivariance error.
The former term embodies the fundamental discrepancy inherently introduced by the specific transformation actions, which naturally exist in the real world and are random.
Hence, in this paper, we assume $\rho$ as learnable permutations that can be modeled as variables following a normal distribution, i.e., $\Delta\sim\mathcal{N}(0,\sigma)$.
These learnable transformations are also inherently norm-conserving, thereby allowing us to implicitly incorporate $\rho$ into our considerations.
Consequently, when an input $x$ exhibits proximity to its transformed version, the outputs under a continuously equivariant function will also maintain close.

The following proposition further shows that the equivariance error of the $\Phi_{\textit{relaxed}}$ will converge to the equivariance error of $\Phi_{gt}\|_{EE}\|$ as they converge in model error.

###### Proposition A.3

Let $\Phi_{gt}\colon X\to Y$ be a function with $\left\|\Phi_{gt}\right\|_{\mathrm{EE}}=\epsilon$. Assume $\left\|\Phi_{gt}-\Phi_{\textit{relaxed}}\right\|_{\infty}\leq c$. Then $\left\|\left\|\Phi_{gt}\right\|_{\mathrm{EE}}-\left\|\Phi_{\textit{relaxed}}
\right\|_{\mathrm{EE}}\right\|\leq 2c+\epsilon.$

Proof.
By triangle inequality and invariance of the norm,

$$ $\displaystyle\left\|g\cdot\Phi_{\textit{relaxed}}(x)-\Phi_{\textit{relaxed}}(g \cdot x)\right\|\leq$ $\displaystyle\left\|g\cdot\Phi_{\textit{relaxed}}(x)-g\cdot\Phi_{gt}(x)\right\|$ $\displaystyle+\left\|g\cdot\Phi_{gt}(x)-\Phi_{gt}(g\cdot x)\right\|+\left\| \Phi_{gt}(g\cdot x)-\Phi_{\textit{relaxed}}(g\cdot x)\right\|$ $\displaystyle\leq$ $\displaystyle 2c+\epsilon.$ $$

$\hfill\square$

###### Definition A.1 (Equivariance Error [ 47 ] )

###### Proposition A.2

###### Proposition A.3

## Peer Reviews

confidence: 5
contribution: 2
initial_rating: 6
presentation: 2
questions: * The paper lacks sufficient exploration of the key parameter b, which controls the perturbation factor Δ, and more extensive experiments with intermediate values of b would strengthen the argument for the necessity of this perturbation parameter.
* Performance deteriorates when (b>0.1), and it would be beneficial to conduct more thorough experiments, especially within the interval [0, 0.1], e.g., 0.02, 0.05, to provide a more definitive analysis of the value of b and its impact on the model’s performance. 
* It would be beneficial to provide concise explanations for abbreviations like SO(2) (Special Orthogonal Group in 2D space) and ER2GCBA (Efficient Relaxed Rotation-Equivariant Group Convolution plus -BA, where the meaning of -BA remains undefined).
* It would be beneficial to investigate the performance of R2Det in larger configurations, such as 'R2Det-L', and compare its performance to the corresponding 'large' versions of YOLO models, to provide a more complete assessment of the model's scalability and potential limitations in handling more complex and computationally demanding tasks.
* It would be beneficial to include a comparative study with more recent and advanced object detection models, such as YOLOv11 and other models, to provide a broader context and demonstrate the model's performance relative to the state-of-the-art.
* This discrepancy raises a question regarding potential differences in the implementation of SRE versus the b=0 setting, or if it could be due to variations in the experimental runs. The authors may please comment on this interesting observation and clarify the reasons behind the difference in AP scores.
soundness: 2
strengths: This work makes a valuable contribution to the field of object detection by addressing the limitations of traditional, strictly rotation-equivariant models and exploring the potential of RRE through ER2GConv. 
This work presents a significant contribution to the field of object detection by addressing a crucial limitation in handling real-world scenarios, where strict rotational symmetries are rarely observed. The authors introduce a novel approach, Relaxed Rotation-Equivariance (RRE), which effectively addresses this limitation. The proposed R2Det model leverages RRE to achieve remarkable performance with a significantly reduced parameter count compared to other leading models. This showcases the model's efficiency and its ability to achieve high accuracy while requiring fewer computational resources. The paper further strengthens its arguments with a rigorous mathematical framework, providing theoretical underpinnings for RRE's effectiveness. Moreover, the plug-and-play nature of the proposed ER2GConv layer allows for seamless integration into existing object detection models, making it a versatile and readily applicable technique. This combination of novelty, theoretical soundness, efficiency, and integration capabilities makes this research highly valuable for advancing the field of object detection.
However, the paper lacks sufficient exploration of the key parameter b, which controls the perturbation factor Δ. While the paper mentions that b=0.1 yields the best results, more extensive experiments with intermediate values of b would strengthen the argument for the necessity of this perturbation parameter and provide a deeper understanding of its influence on performance. Additionally, it would be beneficial to investigate the performance of R2Det in larger configurations, such as “R2Det-L”, and compare its performance to the corresponding “Large” versions of YOLO models. This would provide a more complete assessment of the model's scalability and potential limitations in handling more complex and computationally demanding tasks. The clarity and persuasiveness of the paper can be further enhanced by addressing these specific concerns, and by providing concise explanations for abbreviations like SO(2) (Special Orthogonal Group in 2D space) and ER2GCBA (Efficient Relaxed Rotation-Equivariant Group Convolution plus -BA, where -BA likely refers to a specific architectural component or technique).

* This work makes a valuable contribution to the field of object detection by addressing the limitations of traditional, strictly rotation-equivariant models and exploring the potential of RRE through ER2GConv.
* The authors introduce a novel approach, Relaxed Rotation-Equivariance (RRE), which effectively addresses the limitation of handling real-world scenarios where strict rotational symmetries are rarely observed.
* The proposed R2Det model leverages RRE to achieve remarkable performance with a significantly reduced parameter count compared to other leading models, showcasing the model's efficiency and ability to achieve high accuracy while requiring fewer computational resources.
* The paper further strengthens its arguments with a rigorous mathematical framework, providing theoretical underpinnings for RRE's effectiveness.
* The plug-and-play nature of the proposed ER2GConv layer allows for seamless integration into existing object detection models, making it a versatile and readily applicable technique.
summary: This paper introduces R2Det, a novel object detection model that explores Relaxed Rotation-Equivariance (RRE) to handle real-world scenarios where strict rotational symmetries are often violated. RRE is incorporated into group convolution by introducing the Relaxed Rotation-Equivariant Filter (R2EFilter) and the Relaxed Rotation-Equivariant Group Convolution (R2GConv). The paper further proposes R2Net for image feature extraction and R2Det for 2D object detection, achieving improved convergence and performance with fewer parameters.
weaknesses: * The paper lacks sufficient exploration of the key parameter b, which controls the perturbation factor Δ, and more extensive experiments with intermediate values of b would strengthen the argument for the necessity of this perturbation parameter.
** In page 8, Figure 4(a) and Table 1, the results presented demonstrate a minimal improvement in AP when (b=0.1) compared to (b=0). 
** Performance deteriorates when (b>0.1), and it would be beneficial to conduct more thorough experiments, especially within the interval [0, 0.1], e.g., 0.02, 0.05, to provide a more definitive analysis of the value of b and its impact on the model’s performance. 
* It would be beneficial to provide concise explanations for abbreviations like SO(2) (Special Orthogonal Group in 2D space) and ER2GCBA (Efficient Relaxed Rotation-Equivariant Group Convolution plus -BA, where the meaning of -BA remains undefined).
* It would be beneficial to investigate the performance of R2Det in larger configurations, such as 'R2Det-L', and compare its performance to the corresponding 'large' versions of YOLO models, to provide a more complete assessment of the model's scalability and potential limitations in handling more complex and computationally demanding tasks.
* It would be beneficial to include a comparative study with more recent and advanced object detection models, such as YOLOv11 and other models, to provide a broader context and demonstrate the model's performance relative to the state-of-the-art.
* I noticed an interesting discrepancy in the results presented on page 8, specifically in Table 1 and Table 2. While both tables use the VOC test dataset, the reported AP scores for the SRE (b=0) model in Table 1 (C4) differ from the reported SRE scores in Table 2 (C4).
** In Table 1, the AP50(%) and AP50:95(%) for b=0 are 83.8 and 64.4 respectively, whereas in Table 2, the SRE AP50(%) and AP50:95(%) for C4 are 82.9 and 64.2.
** This discrepancy raises a question regarding potential differences in the implementation of SRE versus the b=0 setting, or if it could be due to variations in the experimental runs. The authors may please comment on this interesting observation and clarify the reasons behind the difference in AP scores.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. In Figure 1 on the left, the input in the lower left corner mapped to the output in the upper right corner should use the relaxed rotation-equivariant function.
soundness: 3
strengths: 1. The paper introduces a novel relaxed rotation-equivariant group convolution (R2GConv), which extends existing equivariant neural networks (ENNs). Additionally, the resulting model, R2Det, shows strong performance across various datasets and tasks.

2. The authors enhance the R2GConv module by incorporating depth-wise and point-wise convolution, and conduct extensive comparative and ablation experiments to confirm its positive impact on the outcomes.

3. The paper is well-written and easy to understand.
summary: This paper points out that symmetry breaking often occurs in the real world. However, traditional GConv-based methods are limited by strict operational rules in group space, ensuring strict equivariance of features only under a limited set of group transformations, making them difficult to adapt. The paper defines the relaxed rotation-equivariant group R4 based on the strict rotation-equivariant group C4 and proposes the relaxed rotation GConv (R2GConv). The paper constructs R2Det using GConv and its derived convolutional structures, achieving excellent results on the PASCAL VOC and MS COCO 2017 datasets, and also verifies the good performance of R2Det in classification and segmentation tasks.
weaknesses: 1. Dataset Limitation: The selected datasets, PASCAL VOC and MS COCO 2017, do not emphasize rotation characteristics, which reduces the impact and relevance of the experimental results. To better highlight the effects of SRE and RRE modeling, rotation-specific object detection datasets should be used.

2. Insufficient Baseline Comparison: it would be beneficial to include comparisons with established models in rotation object detection, such as ReDet and FRED, to strengthen the evaluation and provide more convincing evidence.

---

confidence: 3
contribution: 3
initial_rating: 8
presentation: 3
questions: Suggestions:
- (Weaknesses 1.): The reviewer suggests that the authors fix the mistakes in the preliminary section, and rename the methods and claims accordingly.
- (Weaknesses 2.): The reviewer suggests the authors to fix typos, to reduce the number of acronyms in plain text (e.g., define only R2GConv and call its variants "point-wise R2GConv", "efficient R2GConv", ...), and to only include *simple* and *intuitive* figures in the main article to simplify the reading. The more detailed diagrams could then be moved to appendix.

Questions:
- The performance gap between different $n$-norder cyclic rotation groups is substantial (Table 3). What is the reason for such a gap? Is it only due to the "newer equivariant angles"? How would the introduced architecture perform without equivariance? Also, is there an intuitive explanation for why it is beneficial for 2D object detection (with non-rotating bounding boxes, like on COCO) to use $C_8$ instead of $C_4$, despite the output (bounding boxes) only having symmetries in the $C_4$ group?
- Experiments showed that on both PASCAL VOC and COCO, performance with strict equivariance is inferior to that with the approximate equivariance introduced in the paper. For intuition, could the authors provide specific (intuitive or experimental) examples of 2D object detection in which it is beneficial to break the strict rotation equivariance?

I am willing to raise my rating if my concerns are addressed.
soundness: 4
strengths: - The method demonstrates impressive performance-compute trade-offs on the object detection task.
- The authors evaluated the effect of the main contribution (i.e., the learnable perturbation) on performance. Furthermore, its effect on the learned features is well illustrated in Figure 5.
summary: The paper first discusses a limitation of strict equivariance for real-world applications (i.e. symmetry breaking) and then introduces building blocks and architectures for relaxed rotation equivariance. It focuses mostly on 2D object detection.
More specifically, the authors introduce a relaxation of the group operations on the $n$-order cyclic rotation group $C_n$ by adding learnable perturbations to existing (strict) rotation equivariant filters. They then design different relaxed rotation-equivariant group convolutional modules that serve as building blocks for two architectures: the relaxed rotation-equivariant network (R2Net) and object detector (R2Det).

Experimentally, the paper compares different perturbation levels and shows that a small perturbation leads to stronger performance than strict equivariance (i.e., with no perturbation) on 2D detection datasets (PASCAL VOC and COCO). The new method is also compared to the YOLO series detectors and achieves much higher performance-compute trade-offs. Finally, the R2Net and R2Det methods are evaluated respectively on image classification and instance segmentation.
weaknesses: 1. The provided background and naming of methods are misleading and partially incorrect. Indeed, **relaxed** equivariance is defined in (Kaba and Ravanbakhsh, 2023) as a relaxation that allows breaking the symmetry of inputs and mapping to arbitrary orbit types when necessary. Note that the output of the function is still predictable under the transformation of the input. While in R2Det, the relaxed equivariance is mistakenly defined (definition 1, line 161) using the definition of **$\epsilon$-approximate** equivariance (despite referring to the definition from (Wang et al., 2022a) which correctly names it **$\epsilon$-approximate** equivariance). 
Furthermore, the introduced filter is called a "relaxed rotation-equivariant filter" but is implemented by allowing for some learnable perturbation, which is therefore **NOT** a **relaxed** rotation-equivariance module. Figure 1a also incorrectly illustrates the problem being tackled in the paper. Notably, relaxed equivariance was introduced as an alternative to noise-injection methods: "offering an alternative to the noise-injection methods" (see the abstract from Kaba and Ravanbakhsh, 2023).
The above-mentioned problems make the paper's claims incorrect and could lead to important misunderstandings of already established concepts. 
2. The paper is not easy to read, mainly because of: the abundant use of abbreviations and acronyms (ENN, GConv, RRE, SRE, NRE, R2Filter, R2Lift, R2GConv, DR2GConv, PR2GConv, ER2GConv, ER2GCBA, ...), multiple typos  ("Rotationa-Equivariant" line 25, "we further exploring" line 62, "More analysis can refer to" line 167, "converge when 66-epoch" and  "converges at about 198 in the epoch" lines 401-402, ...) and over-loaded illustrations (e.g., Figure 2 and 3).

Meta-review: The paper introduces Relaxed Rotation-Equivariant GConv (R2GConv), addressing the limitations of traditional GConv models in handling symmetry-breaking, particularly rotational symmetry-breaking. Traditional GConv assumes strict equivariance, which fails to account for real-world deviations in symmetry. The proposed R2GConv relaxes the rotational transformations, introducing only minimal additional parameters.The authors apply R2GConv in the Relaxed Rotation-Equivariant Network (R2Net), and develop the R2Det object detector for 2D detection tasks. Experimental results demonstrate that R2GConv improves natural image classification, while R2Det achieves strong performance in 2D object detection with better generalization and robustness under symmetry-breaking conditions.

All reviews agree to accept this paper. The authors are required to update this paper when preparing the final version, considering the valuable reviews during the rebuttal period.

Decision: Accept (Poster)