R2Det: Exploring Relaxed Rotation Equivariance in 2D Object Detection
Authors: Zhiqiang Wu
Yingjie Liu
Hanlin Dong
Xuan Tang
Jian Yang
Bo Jin
Mingsong Chen
Xian Wei
Venue: ICLR 2025 Poster (2025)
Keywords: Relaxation
Rotation
Equivariance
Primary area: applications to computer vision, audio, language, and other modalities

Abstract: Group Equivariant Convolution (GConv) empowers models to explore underlying symmetry in data, improving performance. However, real-world scenarios often deviate from ideal symmetric systems caused by physical permutation, characterized by non-trivial actions of a symmetry group, resulting in asymmetries that affect the outputs, a phenomenon known as Symmetry Breaking. Traditional GConv-based methods are constrained by rigid operational rules within group space, assuming data remains strictly symmetry after limited group transformations. This limitation makes it difficult to adapt to Symmetry-Breaking and non-rigid transformations. Motivated by this, we mainly focus on a common scenario: Rotational Symmetry-Breaking. By relaxing strict group transformations within Strict Rotation-Equivariant group $\mathbf{C}_n$, we redefine a Relaxed Rotation-Equivariant group $\mathbf{R}_n$ and introduce a novel Relaxed Rotation-Equivariant GConv (R2GConv) with only a minimal increase of $4n$ parameters compared to GConv. Based on R2GConv, we propose a Relaxed Rotation-Equivariant Network (R2Net) as the backbone and develop a Relaxed Rotation-Equivariant Object Detector (R2Det) for 2D object detection. Experimental results demonstrate the effectiveness of the proposed R2GConv in natural image classification, and R2Det achieves excellent performance in 2D object detection with improved generalization capabilities and robustness. The code is available in \texttt{https://github.com/wuer5/r2det}.

Published as a conference paper at ICLR 2025
R2DET: EXPLORING RELAXED ROTATION EQUIVARI- ANCE IN 2D OBJECT DETECTION
Zhiqiang Wu'n Yingjie Liu'
Hanlin Dong'
Xuan Tang²
Jian Yang*
Bo Jin
Mingsong Chen' Xian Weilf
Software Engineering Institute, East China Normal University
2 School of Communication and Electronic Engineering. East China Normal University
3 School of Geospatial Information, Information Engineering University
4 School of Computer Science and Technology, Tongji University
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y yjLiu 33Liu6@hottn weah6mthhtmallllmmmmmmm comm xian, weietum, de
ABSTRACT
Group Equivariant Convolution (GConv) empowers models explore underlying symmetry in data, improving performance. However, real-world scenarios often deviate from ideal symmetric systems caused by physical permutation, character- ized by non-trivial actions of symmetry group, resulting in asymmetries that affect the outputs, a phenomenon known as Symmetry Breaking. Traditional GConv-based methods are constrained by rigid operational rules within group space, assuming data remains strictly symmetry after limited group transforma- tions. This limitation makes it difficult adapt to Symmetry- Brrekkng and non- rigid transformations. Motivated by this, we mainly focus on a common scenario: Rotational Symmetry-Breaking By relaxing strict group transformations within Striet Fouution group C.. we redefine a Relaxed Rotation-EEqivariin group R., and introduce a novel Relaxed Iootation-Eqivaann with only minimal increase of 4n parameters compared to GConv. Based on R2GConv, we propose Relaxed Rotation- Eaquiion-quivariant Network (R2Net) as the backbone and develop a Relaxed Notation-Equivanl for 2D object detection. Experimental results demonstrate the effectiveness of the proposed R2GConv in natural image classification, and R2Det achieves excellent performance in 2D object detection with improved generalization capabilities and robustness. The code is available in https: / /github...... com/ 5 / 2det.
GConv (R2GConv)
Object Detector (R2Det)
INTRODUCTION
2D object detection is crucial computer vision task with applications in various domains, including autonomous driving and geosciences (Zou et al., 2023; Kaur & Singh, 2023). Recent advancements in Deep Neural Networks (DNNs) (Hussain, 2023; Huang et al., 2017: Zagoruyko & Komodakis, 2016; Xie et al., 2017) have achieved remarkable progress. Nevertheless, objects within natural im- ages often exhibit rotation and scale variations, requiring DNNs handle geometric transformations more flexibly. One effective approach to address this issue is data augmentation, which improves object detection performance by rotating the dataset expand additional data volume. Still, leads to considerable memory and training cost. Equivariant Neural Networks (ENNs) (Gerken et al., 2023) enhance feature learning and improve downstream task performance by incorporating symmetry and leveraging underlying physics. For instance, the Strict Rotation-Equiiee Equivariance (SRE) introduced in Group Equivariant Convolution (GConv) (Cohen & Welling, 201 6a) has demonstrated superior performance compared to traditional detectors (Han al., 2021).
Using symmetry as an inductive bias in ENNs has emerged as a powerful tool, with significant conceptual and practical breakthroughs (Bronstein et al., 2021). Note that the symmetry refers to the fact that the input remains the same after transformation, rather than the observed symmetry of the object itself Weyl (1952). However, these ENNs merely or by default assume strict symmetry
"qqual Contribution "oorreponndig Author

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Published as a conference paper at ICLR 2025
across all features after equivariant transformations. Also, real-world data features rarely conform to strict symmetry but often conform to non-strict symmetry after some transformations. Such non- strict symmetry is called Sommetry-Breakkinn which means that features tend to shift from high symmetry to low symmetry in the real world.
As stated in Gross (1996), "The secret of nature is symmetry, but much of the texture of the world is due to mechanisms of symmetry breaking."
Based such natural law, raises a critical question in ENNs follows:
Are existing ENNs considering prior knowledge of such non-strict symmetry in real-world sce- narios, such as 2D object detection and other computer vision tasks?
The answer is No. Following Curie's principle (Earman, 2004; Chalmers, 1970), equivariant func- tions preserve or enhance the symmetry of the input, making them inherently stricter. As result, ENNs are unable to fully align with or adapt to non-strict symmetry present in real-world scenarios.
As stated in Curie's principle, the symmetry of cause is always preserved in its effects, and any asymmetry in the effects must be present in their causes. This implies that the input with higher symmetry cannot be mapped to the output with lower symmetry since the input has no corresponding asymmetry. Conversely, the input with lower symmetry can be mapped to the output with higher symmetry. When an object experiences events such as rotations that exceed the predefined group's ' scope or the introduction of minor defects on its surface, it implies that the training images and the target object may have different levels of symmetry. difficulties in accurately representing the object, 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 (Locatello et al., 2020; Smidt et al., 2021; Wang et al., 2022a; Kaba & Ravanbakhsh, 2023; Huang et al., 2024; Xie & Smidt, 2024) discussed relaxation of equivariance and claimed that relaxed ENNs can model Symmetry- -Beekking in multiple domains.
such cases, strict ENNs may encounter
However, there is still a significant gap between these existing relaxed ENN works and practical applications of the computer vision field, e.g., 2D object detection.
This work especially focuses on a common equivariance in ENNs, i.c., Rotation- Dattiion-Euivvriinncee to Differing from traditional SRE GConv, this work presents
address Rational novel Relaxed Intation-Equivarianee (RRE) GConv by incorporating a learnable adaptive rotational deviation parameter that is updated end-to-end by the dataset. Figure details case of Rotational Symmetry- Bermetty-Breakinn and the proposed approach to solve it. RRE enhances the network's 's ability to recognize objects with relaxed rotational equivariance, effectively tackling the Symmetry-Breaking problem and capturing distinct features from perturbations, thus promoting performance. The main contributions of this work are as follows:
We introduce RRE into the group convolution operation by simply incorporating learnalee pertur- bations. proposing Relaxed Rotation Rotation-Equivaricn Equivariant GConv (R2GConv).
To our knowledge, we are the first explore Rotational Symmetry- Breekking-rrnggggg situations within vision tasks. We further propose a Relaxed Cotation-Equivarrin Network (R2Net) as a backbone for better representation learning for Rotational Symmetry Symmeetry-Breaking.
We redesign a Relaxed Ootation-Equivariaan Object Detector (R2Det) for 2D object detection. Experimental results demonstrate that the proposed approach has achieved better convergence and outstanding performance with lightweight parameters.
2 RELATED WORKS
Conv-based 2D Object Detectors. 2D object detection (Chen et al., 2023; Zou et al., 2023; Kaur & Singh, 2023) is fusion of object location and classification tasks, which involves locating ob- jects through bounding boxes and identifying their respective categories. Several notable methods (Girshick et al., 2014; Liu et al., 2016; Carion et al., 2020; Wang et al., 2021; 2022b) have been developed, and the YOLO framework has stood out for its remarkable balance of speed and ac- curacy (Redmon et al., 2016; Hussain, 2023; Terven et al., 2023). Since its inception, the YOLO family has evolved through multiple iterations and variations, such as YOLOv8 (Jocher et al., 2023) YOLOv9 ( (Wang& et al., 2024b), YOLOv10 (Wang et al., 2024a).
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The ideal feature of a circle with higher symmetry rarely occurs real-world scenarios. Instead, the physical perturbation of higher symmetry results in lower symmetry. While ENNs can handle higher symmetry, Curic's principle dictates features of higher symmetry cannot be mapped to outputs with lower symmetry, inducing feature learning. RRE function ΦRRE can solve Rotational Symmetry- Hemmetry-Breaking situations, which has been proven by Kaba & Ravanbakhsh (2023). Note that Sym(-) denotes the level of its symmetry. Right: This work proposes the R2EFilter to build RRE, by incorporating learnable perturbation A. We show the forward and backward processes of R2EFilter based on relaxed C4. named R4.
Figure Left:
Breeakii situations, and impairing
Symmetry vs. Equivariance. Symmetry is fundamental characteristic where an object remains unchanged after certain transformations. Define the function Sym(-) (-) denotes the level of its symme- try, and the equivariant network Øeq. we have Sym(0e((z)) 2 Sym(x) based Curie's principle. That means equivariance can preserve or enhance the symmetry of the input. In effect, equivariance can be used symmetry compiler to find missing -implle data.
Common Equivariance: Rotation- Rotation-Equivariane Equivariance. The concept of an equivariant network was proposed in Cohen & Welling (2016a), named Group Equivariant Convolution Neural Networks (G-CNN). Kotation-Equivariin
convolution or full connect layer (Li et al., 2018; Marcos et al., 2017: Cohen & Welling, 201 6a; Finzi et al., 2020) guarantees the rotation-equivaaraacee of extracted features under the group operations by a higher degree of weight sharing. Moreover, Rotation- Equivariance has recently become a strongly desired prior bias in object detection tasks. Han al. (2021) propose rotation- equivariant 2D object detector (ReDet) to predict the orientation of aircraft accurately. Most recently. Lee et al. (2024) (FRED) achieves fully Cooaatin-Eqqivariin object detection and enables more genuine non-axis-aligned learning. Wang et al. (2023) (DuFqNet) improves 3D object detection performance by constructing a dual-layer object detection network for 3D point clouds with rotation invariance and extracting local-global invariance features. Although Rotation- Euuiiiiinnuvv has been considered, there is a naive assumption of uniform strict symmetry across all features, neglecting scenarios that require relaxation of equivariant constraints.
oriented
Symmetry- Bemmetry-Breaking vs. Relaxed Equivariance. Physical laws are governed by numerous sym- metries and real-world data, such as complex datasets and graphs. However, it often deviates from strict mathematical symmetry due to noisy or incomplete data or inherent Symmetry- Brreuking-reangg fea- tures in the underlying system. Several works aim break the symmetry. Wang et al. (2022a) investigates 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 & Ravanbakhsh (2023) proposes a novel theoretical guidance for constructing relaxed equivariant multilayer perceptrons, going beyond the straightforward ap- proach of adding noise to inputs and using an ENN (Locatello et al., 2020). Huang et al. (2024) tackles graph symmetry in real-world data by leveraging graph coarsening to establish approximate symmetries and propose bias-variance tradeoff formula based on symmetry group selection. Xie & Smidt (2024) introduces Symmettry-Breakinn parameters sampled as model inputs from a set de- termined solely by input and output symmetrics. These works further observe that breaking more symmetry than needed is beneficial.
In nutshell, real-world data rarely conforms to strict symmetry. Diverse objects inherently possess levels of relaxed symmetry, regardless of modality, e.g., 2D images or 3D vision data (Kaba & Ravanbakhsh, 2023). Consequently, existing methods fail to tackle downstream task performance. To our knowledge, we are the first to address these limitations in the 2D object detection field by adopting the Rotational
affecting
and RRE perspective.
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3 PROPOSED METHOD
3.1 PRELIMINARY
Definition 1 (Strict Equivariance). A learning function P X - Y that sends elements from imput space X to output space Y satisfies Striet Equivariance to group G if Vg. E GGx X there exists px- - GL (X) and py : - GL (Y) actions of G such that @stritt (px(g) x) = pr(g) Ostrice (x),
where GL(-) is general linear group over the space.
Definition 2 (Relaxed Equivariance). (Kaba & Ravanbakhsh, 2023) A learning function Orelaved X Y that sends elements from input space X to oulput space Y satisfies Relaxed Equivariance to group G if Vgi. x G x X there exists E Gxi PX :G GL (X) and py G - GY) actions of G such that
Oretaxed (px (g1) x) = pg Drelaxed (x),
Striet Equivariant Neural Networks. Learning equivariant features is an optimization process for a series of Ostrier function sets in the model. However, the challenge of strict equivariant networks lies in designing trainable 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 (Cohen & Welling, 2016b), respectively.
Relaxed Equivariant Neural Networks. The existing equivariant networks assume that the data completely symmetric. This network approximates strict invariant equivariant function under given group actions. For example, in G-CNN, the shared convolution filters achieve equivariant images at 0, 90, 180, and 270 degrees under the strict constraint of C.. However, real-world data is rarely symmetric. This seriously hinders the potential application of equivariant networks. To solve this problem, in Elsayed et al. (2020) and Wang et al. (2022a), relaxing weight constraints can significantly improve the performance and generalization ability of the model. This work relaxes strict group constraints for relaxed equivariant neural network.
3.2 RELAXED ROTATIONN FILTER (R2EFILTER)
In the following, denote all i {0, 1......... 1} for convenient presentation. We first introduce Relaxed Cotation-Equivariin Filter (R2EFilter) denoted by F. serves as the crucial component of R2GConv. To relax strict group constraints, R2HFilter embeds learnalee parameter A to perturb the group operation based on the n-order cyclic rotation group Cn = (c' = 0, 1, 1}, discrete and finite subgroup of SO(2), which an infinite group that contains set of all dimensional rotation angles. The powers of c' indicates performing rotation operation on the input by degrees, times. The affine matrix A on C, can be defined as follows:
cos (2mí/n) A
- sin (2ri/n)
sin (2ri/n)
cos (osri//)))
Further, let the learnable perturbation factor A E max2x2 u(-b, b), where denotes Uniform distribution with boundary value 6. Our original intention is to provide an appropriate level of initial perturbation to fit A a suitable value. In Section 4, we conduct several experiments with different b from 0 to 0.8, where the experimental results show that a small value of b enhances the model's performance, whereas large value of b detrimentally affects the model's 's performance.
[An. A,2, Ais, Aas) and a Relaxed Rotation Equiiiaiinni group R,, (r') based on Cn and a transformation function T: C, - R,, We consider as an addition, i.c., adding the learnable perturbation factor A, the affine matrix A,, Thus, the perturbed affine matrix A on R,, defined as A = T(A A,) = A + A, where other operations are also available for T, such multiplication, and linear transformation, but not limited
Then, we can define A {A,},A,
Here, we explain the affine transformation from the perspective of coordinate changes. Specifically, given W as Kaiming -nitiiiiiitiiiiiaiiinn (He et al., 2015) 2D convolution weight, and F denotes the transformed filter on R,, Consider this example, let a function CoorSet(-) denote a set of 2D coordinates of the elements in where the coordinate system is at the center of For all coordinates
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Iu 2) CoorSet(W), we have two affine transformations: c'(lu ol) A [u w] on Cn and R, respectively. The transformed coordinates [ūt ü,] for F, are as follows:
Aç . [u el and r' r'u v])
r(uu v)) ([ cos (2mi/n + De -sin (2ri//o n)) + Avz [u]
[ 6.]
(4)
sin (2mi/n) + Ais
cos (2mi/n) + As
where I-] denotes the round down operation for +. Further, if [ūs T.] E CoorSet(F,) we set Fil... ........ Mgg 0.] := W[... u, 2/. For coordinates [ú,, úg]] in F; that remain unassigned after mapping from W, we employ the Bilinear Interpolation method to fill F.(-... ūy dz). Then, we have the complete R2FFilter:
(F.). The construction process of F is detailed in Algorithm In the rest of this work, this process is abstracted by the mapping function N,, i.e., F = (V()).
Algorithm Build R2EFilter based on Cn.
Input: The group order n of C,; The boundary value of Uniform distribution 4(-b,, b); The Initialization W.
Kaiming-Initializattinn
Output: The R2EFilter based on C,:: F.
Initialize F
0
Empty F store compelete R2EFilter >> Learnable perturbation factor with Uniform distribution Get the tensor shape >> Get the output channels Loop the group order n
Initialize A n + ((b,,b)
4 Pett.Tensor_Shape (W)
c - Get Suutput.Channels (W) for i = (n 1) do
cos (os(2Ti///) + Da
As cos(2ti/n) + Ai
0]
A;
Get the perturbed affine matrix
sin(2 + Ass
A Repeat_Iensor cA
> Repeat A for times P Generate affine grid G Sampling on G. to get the i-order filter F;
G
Affine.Grid (A, f
F
<
Grid-Sample (W, G:)
Update F F FUF;
end for
return
3.3 RELAXED ROTATION-EQQIVARIANN GCONV (R2GCONV)
So far, by relaxing strict constraints of group operations on C, through learnable perturbation factor A, we have defined the Relaxed Rotation Equiivrranti group R,, and achieved R2EFilter. Based on GConv (R2GConv), encompassing three variants:
it, we introduce a Relaxed Conation-Eqivante Lifting R2GConv, Point-wise R2GConv, and Depth-wise R2GConv.
Specifically, since the input image and traditional convolution filters are on the plane, Lifting R2GConv designed as an equivariant transformation to replace conventional translation opera- tions. It is applied in the first layer to convert the input data into feature maps on specific group (e.g.. R., in this work), enabling more complex and structured feature extraction. For subsequent layers, filters of wise R2GConv and Depth-wise R2GConv are defined on R,,, facilitating the cquivariant transformation and processing of these feature maps. The point- and depth-wise strategies in R2GConv avoid the inevitably substantial parameters and computational overhead as- sociated with GConv operations. Meanwhile, the reduced number of parameters also diminishes the computational complexity and required time to build their filters with R2EFilter. Based this rea- son, we combine Point-wise R2GConv and Depthwise R2GConv to propose an Efficient R2GConv.
For convenience, in the following, let nı, et, C1+1, hr hrs and wi denote the group order, the number of input channels, the number of output channels, convolution kernel size, input height, and input width in the (-layer, respectively.
R2GConv. Consider the input feature map xt E X and an initial convolution weight W, R xxmm xh, where 7 > Traditional convolution computes the output feature map on a plane by performing an inner product between the input feature map and the convolution filter, with the filter shifted by defined step. According Section 3.2, have R2EFilter F; (WW)) R m1 xem, x xk: where 71++1 = 7i n. Thus R2GConv transforms the input feature map æ; the output feature map #1+1 E Re No-1 Xhe+1 xw follows:
2z+ :
Fild.
mland.
Vid E (0.... ,4419411 1}.
(5)
X
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where * denotes vanilla convolution operation. We also prove that our R2GConv is the RRE block, as shown in Appendix 7.10.
Lifting R2GConv. Considering the special i... 1, to lift the imput feature map 21 E R9 x onto R., called Lifting R2GConv. Thus, have its R2EFilter F v (W,)
where R2GConv is employed in the first layer,
X xki, where ny 1, n2 n. Further, the
output feature map 22 E xha xw2 obtained as follows:
22 [2.d
E (0. ,272 1}.
(6)
hg X
R1 X
X1X
Point-wise R2GConv. In the (-layer with / given the input feature map æ E 1 and
an initial convolution weight Wi E IR Xapu x1x1 we have Fi = ((F))) E RR X1mx where n4 Ri+1 n. Since the size of filter Fī is always considered the standard point-wise convolution operation, which is defined as follows:
1, the Point-wise R2GConv is
F 1X1
wilaad Vd E (0. CC-11 - 1}.
hr-1 XW2+1
Depth-wise R2GConv. Given the input feature map E R xen and an initial convolution
weight W E R x x x in the l-layer with / > 1, we have F/ ((W)) E Ronn p x
where ni M1+1 n.. Consequently, Depth-wise R2GConv is defined follows:
[ddasd xD
A (0, , ,0410441 - 1},
(8)
hp+1 XMP-1
1x
Cpty Xhy X
where *n denotes the depth-wise operation with the convolution group number C14174111 which needs to meet q/c1+1 E Z for the correct division of convolution group number.
Efficient R2GConv. Here, we propose an Efficient R2GConv as 21+1 Fi' (F) * æ;)) in the I-layer. where n4 = 7i+1 = n. Note that the point-wise operation is used for channel connection, while the depth-wise operation reduces the high parameters and computation. With these operations, the model's performance can be effectively maintained while achieving parameter reduction.
3.4 THE RELAXED ROTATION- EQUIVARIANT NETWORK (R2NET)
Based R2GCony, further propose Relaxed Rotation -quivaiin Network (R2Net), as shown in Figure 2. R2Net comprises a Lifting R2GConv and the standard four-stage processing used for most backbone networks. In the first layer, we typically project the input tensor our defined group R, while performing 2 downsampling by Lifting R2GConv with stride 2. Then, we input the projected tensor into the four-stage processing. Each stage incorporates an Efficient R2GConv with stride 2 for 2 downsampling, followed by an R2Net Block for feature extraction. In the last stage, we maintain the output channels unchanged compared to the input channels, which mainly reduces the number of parameters. Regarding R2Net Block, we borrow the idea of dividing channels in Res2Net (Gao et al., 2019) to reduce the number of parameters and FLOPs. In the R2Net Block, initially apply Point-wise R2GConv for the channel change and then split the channel. Subse- quently, for each channel after the first one, we aggregate it with the previous channel before feeding it into its corresponding Bottleneck, which is composed of two stacked Efficient R2GConv modules with a residual connection (He et al., 2016) structure to improve the convergence ability.
Here, provide our R2Net with three sizes: R2Net-N, R2Net-S, and R2Net-M, with the suffixes "-N", "-S" and "-M" indicating the Nano size (as introduced above), Small size, and Medium size, respectively. Detailed parameter settings of R2Net be found in Appendix 7.1.
3.5 THE REDESIGNED RELAXED ROTATION- OBJECT DETECTOR (R2DET)
Based on the backbone R2Net, propose novel Relaxed Notation-Equivarian Object Detector (R2Det). R2Det employs the Feature Pyramid Network (FPN) + Path Aggregation Network (PAN) neck architecture, as shown in Figure 3. Specifically, R2Net mainly extracts multi-scale features

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Figure 2: The architecture of R2Net-N the backbone for feature extraction, where #C denotes the number of Bottlenecks in R2Net Blocks based on the channel sizes and varies with the different sizes of R2Net (i.e., R2Det-N S M). Note that all R2GConv including three variants have normalization (BatchNorm) and activation (SiLU) functions, which we do not show in the figure.
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Figure 3: The architecture of R2Det for 2D object detection with a FPN+PAN neck. Here, we only show a simple architectural diagram. For detailed architecture, please refer to Appendix 7.2.
in R2Det. The neck part in R2Det further aggregates information from multiple paths, allowing for enhanced communication between different levels of the feature pyramid. Following three sizes of R2Net, we also provide our R2Det with three sizes: R2Det-N, R2Det-S, and R2Det-M. More detailed parameter settings of R2Det can be found in Appendix 7.1.
EXPERIMENTS
4.1 2D OBJECT DETECTION ON THE PASCAL VOC AND MS COCO DATASETS
To investigate the effectiivnnes of our method, we conduct extensive experiments the MSCOCO 2017 (COCO) and PASCAL VOC07+12 (VOC) datasets. We set b = 0.1, and the RRE (or SRE) is constructed C2 in all experiments if not specified.
Effeet of in Uniform distribution (d(-b, b) on RRE. We explore the impact the boundary value b of uniform distribution (1(-b, b) for the perturbation
model performance, where larger value of b indicates a greater initial perturbation A and has a higher level of RRE in R2GConv. Table demonstrate that R2Det-N with 6 0.1 achieve the best results in both AP50:95 and AP po- outperforming those with initializations of 0 or too large values. Figure 4a further demonstrates the training curves of our R2Det-N (C,) with varying b on VOC dataset. Since the A in R2GConv can be readjusted by end-to-end learning strategy, we suggest that small initial perturbation can better Hommetry-Breakinnn
align A the Rotational provide specific A values in four Efficient R2GConv, as shown in Appendix 7.4. Note that R2Det (RRE) with b = is not equivalent to R2Det (SRE), as with an initial value of still achieves end-to-end updates by gradient descent.
property of natural image datasets. Additionally, we
Convergence analysis. We analyze the convergence of our R2Det and YOLOV8 models during the training process the VOC training dataset, as shown in Figure 4b. Both R2Det-S M converge in about 66 epochs, much earlier than YOLOv8-S / M, which converge in about 138 epochs and 132 epochs, respectively. Furthermore, YOLOv8-N converges in about 198 epochs, while R2Det-N converges in about 132 epochs, the same as YOLOv8-M is worth noting that the convergence curve of R2Det is smoother than YOLOv8, demonstrating a better convergence process of R2Det.
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(a) AP of R2Det with different b
(b) Also comparison of R2Det and YOLOv8.
Figure 4: 50 curves on VOC test dataset, All models train for 200 epochs with the same settings.
Overall, our R2Det models of all three variants (N/S/M M) exhibit not only outstanding performance in 50 but also more stable training process with faster convergence speed compared to YOLOv8 models. The faster convergence could be related to the fact that our R2GConv can extract rich and relaxed equivariant features, enabling earlier learning of these potential features.
Table 1:
Results of R2Det-N (C+) with different boundary value b on VOC test dataset.
Table 2: Ablation experiments of R2Det-N with SRE and RRE (Ours) on VOC test dataset.
Group
Equiv.
(%))
(%))(%)
Params.
FLOPs
b
AP (%
APso-as(9)
C2
GRE
58.4
78.1
L&M
3.2G
0
83,8
64.4
RRE
59.0 (+0.6)
79.4 (+1.3)
1.MM + 0,256K
3.2G
0.1
0.2
84.1
83.5
64.3
65.1
C
RRE
SRE
65.1 (+0.9)
64.2
84.1 (+1.2)
82.9
2.6M + 0,512K
2.6M
3.3G
3,3G
0.4
83.6
64.4
0.6
82.4
62.6
C
SKR
65.5
84.2
4.4M
3.7G
0.8
80.7
59.7
RRE
67.0 (+1.5)
85.4 (+1.2)
4.4M + 1.024K
3.7G
Table 3: Ablation experiments of R2Det-N with SRE and RRE (Ours) on COCO validation dataset.
Group
Equiv:
APtoan(%))
AP*(%%)
APso(%)
APs(%)
AP*(%)
AP,(%)
Params.
FLOPS
SRE
36,2
39.5
$1.0
18.2
39,8
50.7
1.9M
3.8G
36.6 (+0.4)
39.8
(+0.3)
51.5 (+0.5)
19.4 (+1.2)
39.9
(-0.1)
51.5
(+0.8)
1.9M 0.256K
3.8G
SRE
43,2
47.4
58.7
22.6
48.1
58.3
28M
4.0G
RRR
43.7 (+0.5)
47.8
(+0.4)
59.5
(+0.8
24.2 (+1.6)
48.2 (-0.1)
59.3
(+1.0)
288M + 0.512K
4.0G
C3
SRE
46.8
$1.0
26.0
52.7
62.1
4.5M
4.3G
RRB
47.5 (+0.7)
63.7 (+0.5)
52.1
(+).))
27.9 (+1.9)
52.9 (-0.2)
63.4 (.)
4.5M + 1.024K
4.3G
Note that "Equiv." decotes the type of rotation rotationncquiiee equivariant filer used in R2Det, as SRE indicates regular Striet Rotation Equivariant Filter. whereas RRE. means our propased Relaxed Rotation Rotation-Eiqiisane Equivariant Filter (R2EFilee)).
Ablation experiments evaluating RRE SRE on COCO and VOC. We first conduct ablation experiments on COCO and VOC datasets to discuss the effectiveness of RRE, comparing it with SRE, based three groups, including C2, C4. and Cg. As shown in Table and Table 3, R2Det-N (RRE) outperforms R2Det-N (SRE) in all average precision (AP) metrics. These results demonstrate that RRE can effectively overcome the limitations of SRE mentioned above. Meanwhile, compared to R2Det-N (SRE), our approach results in a negligible increase in parameters quantity (Params.) and floating point operations (FLOPs) compared R2Det-N (SRE), improving model performance with almost negligible parameter increase (i.e., 256 / 512 / 1024) on C2 / C2/ Cs.
Inability of naive convolutions with learnable noise to construct RRE. We conduct experiments that incorporate learnable noise into the traditional convolution filters of YOLOV8 to evaluate its impact on convolution filters. As shown in Table 4, simply adding noise to the filters of the original convolution leads to degraded model performance. This suggests that for operators like the original convolution, which lack inherent rotation caationnequivariance properties, simply adding noise does not confer achieving RRE to handle data exhibiting Rotational Syymmetr-Breaking.
Table 4: Ablation experiments of naive convolution filter w/ (w/o) learnable noise of YOLOv8-N on VOC test dataset.
Table 5: Ablation experiments of YOLOv8-N archi- tecture on C₁ on VOC test dataset.
Equiv.
APseaa(%)
APx:(%)
Params.
Type
A%s)(%)
A (9E)
NRE
57.5
78.6
3.3M
w/o naise
78.6
57.5
SRE
62.8
82.7
3.2M
w/ noise
(-5.5)
51.4 (-6.1)
BRE
64.2 (+1.4)
83.6 (40.9)
3.2M
8

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Published as a conference paper at ICLR 2025
Plug-and-play and generalization ability. We show that by simply replacing and plugging, proposed modules mentioned in Section 3.3 can effectively transfer the RRE property to other archi- tectures, such YOLOv8 architecture. Specifically, we replace the first layer Conv of YOLOv8-N with Lifting R2GConv and replace all other intermediate layer Convs with our Efficient R2GConv. From Table 5, YOLOv8-N (RRE) also surpass YOLOv8-N (SRE) and YOLOv8-N (NRE) in both 50:0: and A0 Note that YOLOv8-N (NRE) means the standard YOLOv8-N model. This ex- periment demonstrates that our R2GConv is glug-and-play and still effective other architectures, demonstrating its generalization ability.
Table 6: Comparison with the latest state-of thtte-of-theartt object detectors on COCO validation dataset.
Method
(%
AP (%6)
(%
(P)(%)
aPM((%)
AP, (%)
Params.
FLOPs
YOLOv8-N
37.3
$2.6
40.5
18.6
41.0
$3.5
3.2M
8.7G
YOLOv8-$
44.9
61.8
48.7
26.0
49,9
61.1
11.2M
28.6G
YOLOv8-M
50.2
67.2
54.7
32.3
55.9
66.5
25.9M
78.9G
YOLOv8-L
52.9
69.8
57.5
35.3
58.3
69.8
43.7M
165.2G
YOL0v9-T
38.3
53.1
41.3
2.0M
7.7G
Y
46.8
63,4
50.7
26.6
56.0
64,5
7.1M
26.4G
YOLOv9-M
51.4
68.1
56.1
33.6
57.0
68.0
20.0M
76.3G
YOLOv9-C
53.0
57.8
58.5
69.3
25.3M
102.1G
YOLOv10-N
38.5
41.7
18.9
42.4
54.6
2.3M
6.7G
YOLOv10-S
46.3
63.0
50.4
26.9
51.1
63.7
7.2M
21.60
YOLOv10-M
51.1
68.1
55.8
33.8
56.5
67.0
15.4M
59.1G
YOLOv10-B
52.5
69.6
57.2
35.1
$7.8
68.4
19.1M
92.0G
YOLOv10-L
53.2
70.1
58.0
35.7
58.4
69.4
24.4M
120.3G
R2DeI-N (C.)
43.7
47.8
24.2
2.8M
4.0G
R2Det-S (C.)
50,0
54.6
30.5
55.7
66.2
9.6M
9.0G
R2Det-M (C.)
53.1
70.3
57.9
36.4
58.7
69.6
22.6M
17.3G
Average precision comparison. As detailed Table 6, we compare our proposed R2Det with ad- vanced YOLO-serie detectors on COCO validation dataset. Since both YOLOv8 and R2Det offer //S M variants, each of which is compared individually. Despite having 12.5% 14.3% 12.7% fewer parameters, R2Det achieves significant improvements of 17.2% 11.4% 5.8% in AP50.95. compared to YOLOv8, respectively. Then, we compare R2Det to the latest advanced models. Also on N / S//AA / M three variants, although these models have fewer parameters than R2Det, achieve AP 50:95 improvements by 14.1% 6.8% 3.3% compared to YOLOv9, and 13.5% 8% / 3.9% compared YOLOv10. 10. Moreover, while R2Det-M has approximately half the number of parame- ters of YOLOv8-L. and lower FLOPPs than YOLOv9-C and YOLOv10-L. it consistently outperforms YOLOv8-L, YOLOv9-C, and rOLOv10-B in AP across all IoU thresholds and for objects dif- ferent sizes. In addition, R2Det-M achieves approximate results compared to YOLOv 10-L, but its parameters are fewer. We further conduct experiments on VOC dataset shown in Table In each variant comparison, R2Det exceeds YOLOv8 in APso by 13.1% / 7.6% / 4.6%, for the N, and M variants, respectively. R2Det-M achieves ofte-of-theean performance in AP on VOC with only one-third parameters of YOLOv8-L. Experimental results above show that our R2Det has achieved excellent performance AP with fewer parameters and much lower FLOPs than existing Mlate-of-the-at object detectors. Please refer to Appendix 7.12 for more results and comparisons.
Table 7: Comparison with advanced YOLOv8
models VOC test dataset.
Table 8: Comparison with other models in Top-1 Accuracy (%) on CIFAR-10/ 100 datasets.
Method
APso(%)
Params.
FLOPs
Method
C-10(%)
C-100(%)
Params.
YOLOv8-N
78.6
3.0M
8.IG
YOLOv8-S
81.6
11.1M
28.5G
WideResNet
95.8
79.5
36.5M
YOLOv8-M
83.7
25.9M
78.7G
ResNeXt-29
96.4
82.7
68.1M
YOLOv8-L
86.4
43.6M
164.9G
DenseNet-BC
96.5
82.8
25.6M
YOLOv8-X
86.9
68.1M
257,5G
R2Net-N (C2)
95.8
80.6
0.9M
R2Det-N (C.)
84.1
2.6M
3.3G
R2Net-S (C.)
96.6
82.2
2.8M
R2Det-S (C.)
86.0
9.6M
8.9G
R2Net-M (C.)
97.3
82.7
6.0M
R2Det-M (C.)
87.3
22.6M
17.2G
4.2
ADDITIONAL EXPERIMENTS
Image classification. Classification evaluation assesses the discriminative quality features. We
construct the backbone R2Net incorporating v8Classify Head for image classification tasks, as de-

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Published as a conference paper at ICLR 2025
picted in Figure 3. Table 8 shows the top-1 accuracy of R2Net in three sizes (-N / M) CIFAR-10 100 datasets. R2Net and all outstanding baselines trained from scratch. CIFAR- 10, R2Net-M achieves superior accuracy with significantly fewer parameters than DenseNet-BC, WedeResNet, and ResNeXt. In 100, R2Net-N has only 2.5% of WideResNet's parameters, R2Net-S has 4.1% of ResNeXt-299 parameters, and R2Net-M has 23.4% of DenseNet-BC" param- eters, yet they still achieve competitive accuracy. Note that R2Net effectively balances parameters and accuracy. These results demonstrate the effectiveness of R2GConv. Please refer to Appendix 7.6 for details, and more results of R2Net with different groups (e.g., C2 Cs).
Instance segmentation. Please refer to Appendix 7.6 for detailed results and analysis.
4.3 VISUALIZATION
Visualization of RRE, SRE and NRE. The visualization of RRE (Ours), SRE, and Equivariance (NRE) is shown in Figure 5. We input (a) into R2Det-N (RRE), R2Det-N (SRE), and YOLOv8-N (NRE) obtain the feature maps (b), (c), and (d), respectively. Observing the blue circles in (b), we notice slight differences but the overall feature maps are relaxed rotation- equivariant, showcasing the characteristic of our RRE. Observing the orange circles in (c), we find rotation-equivariann equivariant, presenting SRE. Lastly, observing the red circles (d), we find almost the characteristic of NRE. More visualizations can found in Appendix 7.7.
that the feature maps are strict rotation
Images
RZDet-N feature maps
mage (6) Y O (NRE) festure majs
Figure 5: Visualization of the rotated feature maps in RRE (Ours), SRE, and NRE based C4+
We also present a case of Rotational Symmetry- Breaking on C, in 2D object detection, as shown Figure We stretch tvmonitor with C2 sym- metry, therefore breaking its C4 symmetry. It can be seen that R2Det with SRE RRE achieves similar detection probabilities in the unstretched image. However, in the stretched image, the detec- tion probability decreased by 11% for R2Det with SRE, while R2Det with RRE only decreased by 3%. This example indicates that RRE can better model Rotational Symmetry-Breakee situations.
Visualization of Rotational Symmetry-Breakiee
Deteciinn Preis
Diletion Proh. 0.89 -0.00
0.8 s0.60
(x) 12DDe-NwithhSEE
(d D2De witth (Ouss
Figure 6: An example of Rotational Symmetry Breettr---Breinng on C₁ in 2D object detection.
CONCLUSION
In this work, we propose novel approach build RRE group R., by perturbing a SRE group C,, Based on R,,, we form well-designed R2GConv operation, which tackles Rotational Symmetry- Breaking situations to better align with real-world scenarios. Furthermore, we propose an efficient backbone R2Net and a redesigned 2D object detector named R2Det. Experiments demonstrate that our proposed R2Det achieves ofte--of-hh-ar performance compared to models without symmetry bias or with SRE constraints in 2D object detection. Additionally, R2Net feature extraction network can be extended to more complex visual tasks and scenes, leveraging the advantages of RRE by the proposed R2GConv operation.
10

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Published as a conference paper at ICLR 2025
ACKNOWLEDGMENTS
This research is supported by the General Program of Shanghai Natural Science Founda- tion 1400242 20.232R1419300) the National Natural Science Foundation of China (No.42130112, (0.442371479) the Science and Technology Commission of Shanghai Municipal- ity (No.22D222220004), and Shanghai Frontiers Science Center of Molecule Intelligent Syntheses.
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7 APPENDIX
7.1 MODEL ARCHITECTURE OF R2DET
In this section, we provide details of R2Det-N S M, as shown in Table 9.
Table 9: The parameter settings of R2Det-N / $ M. Among them, the Index column denotes the module index; the From column denotes where the input of this module comes from. For example, denotes the upper module, -1, denotes the upper module, and 6-module; the OS and OC columns denote the output size and channels from the upper module, respectively. In the OC column, n denotes the group order of our defined group R., Note that the channel and size of the first input are 3 and 640 x 640, respectively.
R2Det-N
R2Det-S
R2Det-M
Index
From
Module
OS
OC
Params.
OC
Params.
OC
Params.
0
-1
Lifting R2GConv 320 x 320
16n
480
32n
944
48
1408
1
Efficient R2GConv
160 160
32n
2416
64n
8912
96n
19504
2
R2Net Block
160
32n
9640
64n
37680
131
3
Efficient R2GConv
80 x 80
64n
8912
128n
34192
192n
75856
4
R2Net Block
80 x 80
64n
58784
128n
232192
192m
409664
5
Efficient R2GConv
40 x 40
128n
34192
256n
133904
384n
299152
6
R2Net Block
40 x 40
128m
232192
256n
923072
384n
1630208
7
Efficient R2GConv
20 x 20
128m
66960
256n
2649766
384n
594064
8
R2Net Block
20 x 20
128m
149056
256n
592992
384n
2072704
9
R2SPPF
20 x 20
128m
65792
256n
262656
384m
590592
10
Efficient R2GUp
40 40
64m
33040
128n
131600
192n
295696
11
6
Concat
40 x 40
192m
384n
0
576n
12
-1
R2Net Block
40 40
128m
181824
256n
724064
384n
2515072
13
Efficient R2GUp
80 x 80
64n
33040
128n
131600
192m
295696
14
-1.4
Concat
80 x 80
128r
256n
384zz
15
R2Net Block
80 80
64n
54064
128n
214592
192n
741472
16
-1
Efficient R2GConv
40 x 40
64n
17104
128n
66960
192n
149584
17
-1,
12
Concat
40 x 40
192n
384n
0
576z
18
R2Net Block
40 x 40
128m
181824
256n
724064
384m2
2515072
19
-1
Efficient R2GConv
20 x 20
128n
66960
256n
264976
384n
594064
20
-1.9
Concat
20 x 20
256n
512n
0
768n
21
-1
R2Net Block
20 x 20
256n
592992
512m
2365600
576n
5320864
22
15
Transfer Block
80 x 80
64n
4288
128
16752
192
37408
23
18
Transfer Block
40 x 40
128n
16752
256
66256
384
148528
24
21
Transfer Block
20 x 20
256n
66256
512
263568
576
333376
25
22,
23, 24
v8Detect Head
897664
2147008
3822016
Total Params
2.8M
9.6M
22.6M
FLOPS
4.0G
9.0G
17.3G
7.2 DETAILED ARCHITECTURE OF R2DET AND R2NET.
R2Det leverages the outputs {O2, Oz. O₂} from the last three stages of R2Net as inputs to fuse fea- tures across various scales and semantic levels, thereby allowing R2Det to detect objects of diverse sizes within an image. The output O4 is first refined by an R2SPPF designed for multi-scale spatial max pooling. Then O2, O₂ and the refined O4 are fed into standard FPN+PAN neck part, which includes R2Net Block, Efficient R2GConv, and Efficient R2GUp for 2 upsampling.
Efficient R2GUp adopts the same architecture as Efficient R2GConv but uses the transposed con- volution operation for upsampling during the depth-wise convolution stage. R2SPPF, similar to the architecture of R2Net Block, optimizes for parameter efficiency and enhances feature representation by capturing information various scales.
The obtained features {Ō,, Og. θ2) are then input into Transfer Block for channel reduction, prepar- ing for final detection by a universal v8Detector Head of YOLOv8. Transfer Block, which incorpo- rates Efficient R2GConv, ensures the features appropriately shaped for the subsequent heads.
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v&Detector Head is anchor-free approach for object's Bounding box (Bbox) and Classification (Cls) predictions, simplifying the prediction process.
Chazzel Concateate (Coaat)
ElatecLAni
Objject
beight x scdt
Rihit Blcck
Tranfte Block
Elfideet R2Clip
B3DCCor
Mio Postinn
beight vith 640 640
32N kbtnn)
begft x with 40x 40
R2lvet Block
villebct
Kd
Charnel Concat Polst wise REECCEN
KAGUp
preiicio
[lbod
6c1 R25P99
R2GSPFE
RiNkt Black
Tranfer Block
Shape Redzz
1b) R2Net Classiicatoz Task
Transter Block
vBClasill Hiee
KChi gridition
16 Trindfer Black
Figure 7: Detailed R2Det for 2D object detection task and R2Net for classification task.
7.3 HYPER-PARE SETER-PARAMETER SETTINGS
The syper-parameter settings of R2Det are shown in Table 10. R2Det employs the same hyper- parameter settings of YOLOv8 but only is trained in 300 epochs, which are less than YOLOv8's 500 epochs. And the hyper-parameter settings of R2Net are shown in Table 11.
Table 10: Hlyper-parameee settings of R2Det.
Table 11: sepeerparameter settings of R2Net.
Hyper parameter
Value
Hyper parameter
Value
Epochs
300
Optimizer
SGD
Epochs
200
Initial learning
0.01
Optimizer
SGD
Finish learning rate
0.0001
Initial learning rate
Learning rate decay
linear
Finish learning rate
0.0001
Momentum
0.937
Learning rate decay
linear
Weight decay
0.0005
Momentum
0.937
Warm-up epochs
3
Weight decay
0.0005
Warm-up momentum
0.8
Warm-up epochs
Warm-up bias learning rate
0.1
Warm-up momentum
0.8
Box loss gain
7.5
Warm-up bias learning rate
0.1
Class loss gain
7.5
Class loss gain
7.5
DFL gain
1.5
HSV-Hue augmentation
0.015
HSV-Hue augmentation
0.015
HSV-Saturation augmentation
0.7
Saturattn augmentation
0.7
HSluealue augmentation
HSV- Value augmentation
0.4
Translation augmentation
0.1
Translation augmentation
0.1
Scale augmentation
0.5
Scale augmentation
0.5
Flip left-right augmentation
0.5
Flip left-right augmentation
0.5
Mosaie augmentation
1.0
Mosaic augmentation
1.0
Close mosaic epochs
Close mosaic epochs
10
7.4 ANALYSIS ON THE LEARNABLE PERTURBATION PARAMETER A
The tendency of A is intrinsically linked to the degree of Rotational Hoammetry-Breakinn within the system (or dataset). Here we further present the final values, mean value, and variance from the first four layers of the full-trained Efficient R2GConv in the R2Det-N model in Table 12. Our R2Det
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has effectively integrated Rotational Semmetry-Breakiing prior, with A in the proposed R2GConv capable of being updated end-to-end. These values more accurately reflect Rotational Symmetry- Breaking phenomena in natural datasets, i.e., minor Relaxed Rotaion- Euuiiion-Euvvaiianeee (RRE).
Table 12: The distribution of A values for four Efficient R2GConv in the full-trained R2Det-N on VOC training dataset. All of them are initialized from the Uniform distribution U 0.1,0.1).
4
A2
A
A
Mean value
Variauce
-0.0127
0.0265
-0,0041
0.0325
0.0310
0.0281
-0.0577
-0.0255
-0.0353
0.0114
0,0072
-0.0310
-0,0070
0.0317
0.0024
0.000624
0.0063
0.1191
0.1082
0.0729
0.0629
0.0916
0.1458
0.0432
0.1504
-0.1154
-0.0028
-0.1312
0..1335
0.0094
0.0275
0.0356
4000355
-0,8716
19.0219
-0.1702
0.01111
0.0856
-0.5176
-0.5718
-0.1320
0,2800
0.0681
0.2178
0.5107
0.2120
0.1603
0.0665
-0.138
-0266733
0.32544
-0.21484
0.000070450
0.033966
0,44604
0.45264
-0.02355
0.54473
0.4430
0.00614417
-0.0033
0.45483
0.42903
0.15881
0.18347
0.0922
7.5 ANALYSIS ON YOLOv8-N-CLS AND R2NET-N ON RIN-MI DATASET.
This section compares the training accuracy of N-LLSS-N-CLS and our R2Net-N on the ROT- MINIST dataset, as shown in the following Table. Note that our designed ROT-MINIST different from standard Rotated MINST, which aims to test the robustness of our R2Net Specifically, we manipulate the training set by randomly rotating 60, 000 images by O, 90, 180, and 270 degrees while maintaining 10, 000 images unaltered in the test set to evaluate the performance of model under rotation. In Figure 8, both R2Net-N and YOLOv8-N-CLA display fluctuations during train- ing. However, R2Net-N exhibits milder fluctuations compared to the more pronounced oscillations observed in YOLOv8-N-CLS, This contrast highlights the superior rotation anti-interference capa- bility of R2Net-N, which is primarily attributed its novel Relaxed RotuationEEuuivariinnc (RRE).
Table 13: Comparaison of the robustness of YOLOv8-N-CLS and R2Net-N (C.)).
Method
Dataset
Error
Params.
YOL
MINST
0.58
1.5M
ROT-MINIST
44.88
MINST
0.54
R2NeINN (C.)
0.8M
ROT-MINIST
27.75
trainloss
vallooss
train/
valiloss
22
resuls smooth
2.20
smooth
2.30
2.0
2.05
2-
2-
2.00
1-
2-
1.95
1.8
0
40
20
mettric/ecccuacyccooo top1
metrits/acccuacyy
metrics/eccuracy
metricc/accuracyy
0.35
0.950
0.90
0.6
0.50
0.30
0.900
0.5
0.85
0.875
0.850
0.75
0.825
0.70
0.35
0.000
(e) RZNNN-
(b) YOLL
Figure 8: Comparison of YOLOv8-N and our R2Net-N in accuracy on ROT-MINST dataset. Both
models are trained for 50 epochs with the resized input size 224 x 224 on dual 4090 GPUs
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7.6 ADDITIONAL EXPERIMENTS ON CLASSIFICATION AND INSTANCE SEGMENTATION
Classification. Also, we construct the backbone R2Net in three sizes (i.e., -N / S / M) on three groups (i.e., C2/ / C4 Cs), incorporating the v8Classify Head for image classification, shown in Figure 3. Table 8 shows the classification performance of our R2Net on CIFAR-10/ / 100 datasets.
R2Net-M (C2) and R2Net-S (C₂) both outperform other models in the accuracy of the CIFAR-10 dataset, but their parameters are much fewer. Also, R2Net-S (Cş) and R2Net-M (C&) both exceed other models in accuracy of CIFAR-I00 dataset. Still, they have fewer parameters compared to other models. We found that R2Net-S (C4) balances the arrameter-accurage trade-offs. These experiments show that R2Net still achieves excellent performance in natural classification tasks.
Table 14: Comparison of other models in top-1 accuracy (%) on CIFAR-10/ 100 datasets.
Method
CIFAR-10 (%)
CIFAR- 100 (%)
Params.
WideResNer
95.8
79.5
36.5M
ResNeXt-29
96.4
82.7
68.1M
DenseNet-BC
96.5
82.8
25.6M
R2Net-N (C2)
94.3
77.4
0.6M
R2Net-S (C2)
95.9
79.5
1.7M
R2Net-M (C2)
96.7
80.5
3.4M
R2Net-N (C2)
95.8
80.6
0.9M
R2Net-S (C.)
96.6
82.2
2.8M
R2Net-M (C.)
97.3
82.7
6.0M
R2Net-N (Cx)
96.2
81.5
1.4M
R2Net-S (Ca)
96.9
83.6
5.0M
R2Net-M (C&)
97.7
84.4
11.2M
Table 15: Comparison of other models in instance segmentation on COCO-seg dataset.
Method
Box AP(%)
Box 5050(%) (%)
Mask AP(%)
Mask (%))
Params.
FLOPs
YOLOv8-N-ss
36.2
51.2
29.6
48.2
3.4M
12.6G
YOLOv8-S seggg
44.0
60.4
36.0
56.8
11,8M
42.6G
RTMDet-Ins-
40.5
35.4
5.6M
11.8G
RTMDet-Ins-S
44.0
38.7
10.2M
21.5G
R2Det-N-seg (C4)
43.7
59.3
35.8
56.2
3.0M
7.9G
Note that R2Det-N-seg adopts the same architecture of R2Det-N but replaces v&Detect Head with v&Segment Head.
Instance Segmentation. We also conduct instance segmentation tasks on the COCO-seg dataset, R2Det-N-seg exhibits improvements of 20.7% in Box AP and 20.9% in Mask AP, with fewer parameters. R2Det-N-sse --sgg achieves similar Mask AP of YOLOv8-S-see and RTMDet-Ins-Ting with only 25.4% and 53.6% of their parame- ters. Moreover, when compared to RTMDet-Ins-S, R2Det-N-seg has similar Box AP while utilizing only 29.4% parameters. The results show that our R2Det also applicable instance segmentation.
shown in Table 15. Compared YOLOv8-N-se
7.7 VISUALIZATION ANALYSIS ON RELAXED ROTATION--QQIVARIINCC
In this subsection, we present a visualization of feature maps from our R2Det-N, as illustrated in Figure 9. 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 (RRE) property of our network.
7.8 HEATMAP VISUALIZATION
In this section, we present the visualization of LayerCAM (Jiang et al., 2021) heatmaps derived from YOLOv8-N, YOLOv7, YOLOv5, and our R2Det-N (C4), as depicted in Figure 10. These heatmaps enable us to locate the regions of interest where the network concentrates its attention. It can be seen that YOLOv7 and our R2Det-N achieved better feature focusing. Notably, R2Det-N shows comprehensive focusing range on certain objects, such as dogs and zebras. In contrast, YOLOv8 and 5 fail to exhibit such targeted feature focus these particular objects.
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Original image
(b) Original image by 90-
(e) Feature meps of (a)
(f) Festure maps of (b)
(c) Original ritated by
Original image mitate by 270*
(g) Feature maps of
(h) Feature maps of
Figure 9: The R2Det-N (C4) feature map visualization of the original image rotated at (a) 0, (b) 90, (c) 180, and (d) 270 degrees, depicted in (e), (f), (g), and (h), corresponds its 32 channels.
YOLOv5 Figure 10: LayerCAM heatmaps visualizations of neck networks across YOLO (v8-N, v7, v5)) and our R2Det-N (C4) models. Examples are from the COCO dataset.
Original image
R2Det-N
YOLOv8-N
YOLOv7
7.9 PARAMETER ANALYSIS OF EFFICIENT R2GCONV (OURS) AND GCONV ON C,
Assuming the input, output channels, and kernel size of both Efficient R2GConv and GConv Cins Cout, and k. respectively. The parameters of Efficient R2GConv can be calculated as follows:
Cin X Cout X n x 1 x (Point-wise R2GConv) + Cout X 1 x x l: x & (Depth-wise R2GConv)
+ x 2 x 2 n X Cin X + k2 Cout
where in (:) denotes the source of parameters.
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The parameters of GConv can be calculated as follows:
k x k = n X Gin. X Cout k.
Cin Cout
Therefore, the parameter of our Efficient R2GConv is only
Y. X Gin X Cout + k2 Cout
Ti X Cin X Cout X k.2
122
n Cin
of GConv.
7.10 PROOF OF GCONV (SRE) AND R2GCONV (RRE)
Definition 3 (Approximate Equivariance). (Wang al., 2022a) A learning function Qupprex X Y that sends elements from inpul space X to output space Y satisfies Approximate Equivariance to a group if Vg, x E EGG X there exists px
GL (X) and py : G GGLG) actions of G
such that
Wppppro
x) pr (g) (g)
(xppror(X]]] c:
(9)
where a small с indicates strong equivariance and relatively larger (not very large) exhibits greater relaxation. Especially, Papprox is equivalent to thaia and also satisfies Eq. when e =
In effect, Approximate Equivariance can be seen Relaxed Equivariance since they are defined by different perspectives. Approximate Equivariance is defined from the I.2 Norm, which can be computable. Relaxed Equivariance is defined from the subgroup, which is universally applicable but more abstract. In this paper, we use the definition of Approximate Equivariance to prove the following conclusions.
On C group, define c-). as rotation of by 2xi/n, and c (++1) med (-). We have the following conclusion (note that for simplicity, we ignore the input and output channels):
Conclusion 1. C,,-Coovv (Vanilla Group Convolution) is strict rotation rotation-equivarian equivariiut block.
Proof.
Given input I and initial weight t, we obtain the strict rotation-equivariant cabbb in the i-order of C., Then C,--GCnv can be defined as: (x) = Ei= x. vp = En c*b))... For any j E 1,.,, 1}, we have
filter astriet
-iild E czb))())))))))))))))))))))))))))
-c((f(z)) e' x* c'( Σ (z) * (v)
EED (x) (+t)) mad ((v) -n- (x) c (b))
Therefore, (x) e (z)) According to Eq. complete the proof.
Conclusion C, C,,RR2Gonnv (Ours) is a relaxed rotation-equivanl equivariant block.
Proof. Given input I, initial weight vh an affine transformation function and the learnable perturbation A, we obtain the relaxed rotation eeuiivrrinniv filter virelaxed t' t''',, A) the i-order
C.. Then Cn-RRGConv can be defined as: fa(x) = En x relaaxed Ei x t* A). For any 10,1,.-. .n 1}, we have:
f2(c((x)) EiD ( ((, ((((2))) <E x* A)) Di (
Therefore, (1) = - e² (z))tt A) A)- et (x) (t' A))I =
Σn= (x) (t' A) (t' ()))i < According to Eq. ??, complete the proof.
In particular, when A 0. we have (1 A) c')), thus 11/2(e) (x)) (f2(x)) = 0, i.e.,
E 0, and fe is strict rotation-cou eauiiinrinuvviiiattt when 0.
7.11 THEORETICAL ANALYSIS
Since the existing methods, including strict rotation-equivarian models, cannot perfectly tackle scenarios in object detection tasks, ie.. they can not learn a relaxed equiv- ariant function. We assume from Wang al. (2022a) that the ground truth function, named dgt. is
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relaxed equivariant. Firstly, provide the Equivariance Error (EE), which quantifies how much the ground truth equivariant function dgt deviates from being strictly equivariant. The EE is de- fined as the maximum deviation from the strict-equivariant behavior under transformations, thereby illustrating how we can regulate the level relaxation in the equivariance property.
Definition 7.1 (Equivariance Error) Let thtt: X + Y be function and G be group. Assume that G acts on X and via px: G - GL(X) and pv: - GL.(Y.. For any g, x G x X, the Equivariance Error of Qgt is defined follow:
b EE = sup I (g) (g((x) gg((px(1) x)]l
x.9
where || denote the No rm operation. According to Equation ?? the main text, the function Øot is relaxed (or e-approximate) equivariant if and only if [| I e, where represents the maximum level of deviation from strict equivariance for Φyt to be considered relaxed equivariant.
Next, we prove that if an input is close to its transformed version, the images under continuous relaxed equivariant function still have to be close. This captures the idea of symmetry in the task, meaning that the output of the function is predictable under transformations of the input.
Proposition 7.2 Let Ogt be relaxed (or e-approximate) equivariant and Lipschitz with constant k. Then, we have
(op - 0g) (x]] & kllpx( x x|| +,, Vx.XEGxX.
Proof. If dht is Lipschitz with constant k, we have (e x) - (x)]l = kl)px(g) x- x||, x-x1,
Vg,
Further, from the EE definition and triangle inequality, we have
Ilpx (g) (x) (x)(x < Pl (g) (x)(x) - (p (g) x x)ll + ( (g) x) (x) < kl)lx(() (9) x x|| + c, Xx.X
Note that both terms, xl(px(gg x x|| and c, collectively determine the overall upper limit of the equivariance error. The former term embodies the fundamental discrepancy inherently intro- duced by the specific transformation actions, which naturally exist in the real world and are random. Hence, in this work, we assume P as learnable permutations that can be modeled variables A following Uniform distribution, i.e., A ~ U( b). These learnable transformations are also in- herently norm-conserving, thereby allowing us to incorporate p into our considerations implicitly. Consequently, when an input exhibits proximity to its transformed version, the outputs under a continuously equivariant function will also maintain close. We further provide visualization exper- iments in Seciton 4.3 and Appendix 7.7 as evidence. Moreover, &gt does not require maintaining all the symmetries of the input, in contrast to strict equivariance, which imposes constraints on stabilizer of the output. Finally, since the model &relaxed aims to approximates dot. the following proposition shows that the equivariance error of the &relaxed will converge to the [ogellEE.
Proposition 7.3 Let Ogt: X + Y be function with l = e. Assume ll Oretaxed looc S c. Then pp (g) Preiaxed (x) Orelaxed px (g) x)] 2c
Proof. By triangle inequality and invariance of the L2 L2-Norm, we have
Ipy (g) Dretaned (x) Pretaxed (px (g) x)|| < |ly (0) (retase(()) pr(g) @x((x)]
+ ellyy (g) (g)(x) - 0ge(px (g) x)]
+ +1 (px (g) x) Omiamall (g) xx]l
cctc+c c= tc
Vg.x G X.
7.12 MORE COMPARISON OF STATE-OF-THE-ART OBJECT DETECTORS.
We also provide a more detailed comparison of state-of-the the object detectors, e.g., YOLOV5, YOLOv6, YOLOv7, and Gold YOLO models, shown in Table 16.
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Table 16: Comparison of state-of-the- object detectors on the COCO validation dataset.
Method
AP 50:0000
50%)
(%6)
AP,. (%%
AP M (%)
AP, (%6)
Params.
FLOPs
YOLOv5u-N
34.3
49.7
37.2
16.8
38.1
48.4
2.6M
7.7G
YOLOv5u-S
43.1
59.9
47.2
24.7
47.6
58.4
9.1M
24.0G
YOLOv5u-M
49.1
66.0
53.8
31.2
54.2
25.1M
64.2G
YOLOv5u-L
52.3
69.2
57.2
34.8
57.9
53.2M
135.0G
YOLOv5u-X
53.3
70.2
58.2
36.9
58.9
69.3
97.2M
246.4G
YOLOv6-N v3.0
37.0
52.7
4.7M
11.4G
YOLOv6-S v3.0
44.3
61.2
18.5M
45.3G
YOLOv6-M v3.0
49.1
66.1
34.9M
85.8G
YOLOv6-L v3.0
51.8
69.2
59.6M
150.7G
YOLOv7
51.2
69.7
55.9
31.8
55.5
65.0
36.9M
104,7G
YOLOvT-X
52.9
71.1
51.4
36.9
57.7
68.6
71.3M
189.9G
YOLOv7-N AF
37.6
$3.3
40.6
18.7
41.7
52.8
3.1M
8.7G
YOLOvT-S AF
45.1
61.8
48.9
25.7
50.2
61.2
11.0M
28.1G
YOLOv7
53.0
70.2
57.5
35.8
58.7
68.9
43.6M
130.5G
YOLOv8-N
37.3
52.6
40.5
18.6
41.0
$3.5
3.2M
8.7G
YOLOv8-S
44.9
61.8
48.7
26.0
49.9
61.1
11.2M
28.6G
YOLOv8-M
50.2
67.2
54.7
55.9
25.9M
78.9G
YOLOv8-L
52.9
69.8
57.5
35.3
58.3
69.8
43.7M
165.2G
YOLOv8-X
53.9
71.0
58.7
35.7
59.3
70.7
68.2M
257.8G
Gold YOLO-N
39.6
55.7
19.7
44.1
$7.0
5.6M
12.1G
Gold YOLO-S
45.4
62.5
25.3
50.2
62.6
21,5M
46.0G
Gold YOLO-M
49.8
67.0
32.3
55.3
66.3
41.3M
87.5G
Gold YOLO-L
51.8
68.9
34.1
57.4
68.2
75.1M
151.7G
YOLO MS-N
43.4
60.4
47.6
23.7
48.3
60.3
4,5M
17.4G
YOLO MS-S
46.2
63.7
50.5
26.9
50.5
8.1M
31.2G
YOLO MS
68.6
55.7
22.2M
80.2G
GELAN-S
46.7
63.0
50.7
25.9
51.5
64.0
7.1M
26.4G
GELAN-M
67.9
55.7
56.4
20.0M
76.3G
GELAN-C
52.5
69.5
57.3
35.8
57.6
69.4
25.3M
102.1G
GELAN-E
55.0
71.9
60.0
38.0
60.6
70.9
57.3M
189.0G
YOLOv9-S
46.8
63.4
50.7
26.6
$6.0
64.5
7.1M
26.4G
YOLOv9-M
51.4
68.1
56.1
33.6
57.0
68.0
20.0M
76.3G
YOLOvv-C.
53.0
70.2
57.8
36.2
58.5
69.3
25.3M
102.1G
YOLOx9-E
55.6
72.8
60.6
40.2
61.0
71.4
57.3M
189.0G
YOLOv10-N
38.5
$3.8
41.7
18.9
42,4
$4.6
2,3M
6.7G
YOLOv10-S
46.3
63.0
50.4
26.9
51.1
63.7
7.2M
21.6G
YOLOv10-M
51.1
68.1
55.8
33.8
56.5
15.4M
59.1G
YOLOv10-B
52.5
69.6
57.2
35.1
57.8
19.1M
92.0G
YOLOOv10-L
53.2
70.1
58.0
35.7
58.4
69.4
24.4M
120.3G
YOLOv10-X
54.4
71.4
59.4
37.1
59.9
71.1
29.5M
160.4G
C2-R2De-N
43.7
59.5
47.8
24.2
48.2
59.3
2.8M
4,0G
R22e2
50.0
66.5
54.6
30.5
55.7
66.2
9.6M
9.0G
C.-R2Det-M
53.1
70.3
57.9
36.4
58.7
69.6
22.6M
17.3G
21


Reviews:
[{"review_id": "uT75UJQxdW", "reviewer": "Reviewer_e3oq", "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.", "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.\n* 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. \n* 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).\n* 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.\n* 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.\n* 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.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 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. \nThis 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.\nHowever, 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).\n\n* 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.\n* 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.\n* 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.\n* The paper further strengthens its arguments with a rigorous mathematical framework, providing theoretical underpinnings for RRE's effectiveness.\n* 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.", "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.\n** 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). \n** 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. \n* 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).\n* 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.\n* 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.\n* 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).\n** 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.\n** 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "EjlirDY7AC", "reviewer": "Reviewer_L1nv", "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.", "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.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 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.\n\n2. 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.\n\n3. The paper is well-written and easy to understand.", "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.\n\n2. 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lCYZ2N2zUS", "reviewer": "Reviewer_AppE", "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.\nMore 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).\n\nExperimentally, 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.", "questions": "Suggestions:\n- (Weaknesses 1.): The reviewer suggests that the authors fix the mistakes in the preliminary section, and rename the methods and claims accordingly.\n- (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.\n\nQuestions:\n- 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?\n- 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?\n\nI am willing to raise my rating if my concerns are addressed.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- The method demonstrates impressive performance-compute trade-offs on the object detection task.\n- 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.", "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). \nFurthermore, 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).\nThe above-mentioned problems make the paper's claims incorrect and could lead to important misunderstandings of already established concepts. \n2. 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).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)