Learning Color Equivariant Representations

## Abstract

Abstract In this paper, we introduce group convolutional neural networks (GCNNs) equivariant to color variation.
GCNNs have been designed for a variety of geometric transformations from 2D and 3D rotation groups, to semi-groups such as scale.
Despite the improved interpretability, accuracy and generalizability of these architectures, GCNNs have seen limited application in the context of perceptual quantities.
Notably, the recent CEConv network uses a GCNN to achieve equivariance to hue transformations by convolving input images with a hue rotated RGB filter.
However, this approach leads to invalid RGB values which break equivariance and degrade performance. We resolve these issues with a lifting layer that transforms the input image directly, thereby circumventing the issue of invalid RGB values and improving equivariance error by over three orders of magnitude.
Moreover, we extend the notion of color equivariance to include equivariance to saturation shift.
Our hue-, saturation-, and color-equivariant networks achieve strong generalization to out-of-distribution perceptual variations and improved sample efficiency over conventional architectures.
We demonstrate the utility of our approach on synthetic and real world datasets where we consistently outperform competitive baselines.

## 1 Introduction

The tremendous progress of image classification in the last decade can be readily attributed to the development of deep convolutional neural networks [; ; ].
The highly nonlinear mapping and large parameter space does not lend itself to interpretation easily, but even in early networks, representations of geometry and color were observed and recognized for their importance [; ].
While a relatively large body of literature has worked to improve the robustness of convolutional neural networks to geometric transformations [; ; ; ; ; ], improving their robustness to perceptual variation has garnered considerably less attention.

A commonly used heuristic for improving network robustness to color variation is to perform mean subtraction and normalization on training set examples.
This approach can work well when the training and testing datasets are drawn from the same distribution; however, for data that are collected at different points in time, or with different sensors, this is not likely to be the case.
Consider, for example, the case of medical imaging where images of tissue samples collected from different labs (or from the same lab at different points in time) may have different characteristics due to variability in data collection protocols or imaging processes [].
This variability presents a significant challenge for convolutional neural networks which have been found to be sensitive to color variations even on the level of individual blocks [].
Moreover, when presented with color perturbations, conventional networks exhibit a significant drop in classification performance [].
One approach to mitigate the effect of color variation is to ignore color entirely by enforcing color invariance.
This can be done by converting input images to grayscale, or enforcing representation similarity across color as is done in [].
This approach has its own challenges, however, since in many domains, color is an important cue for classification.
Another approach is to use dataset augmentation, a technique that can improve robustness in the presence of known transformations [].
This approach, however, requires extended training times and does not provide improved interpretability.

In this work, we address the challenge of neural network sensitivity to color variation by leveraging the geometric structure of color in a group convolutional neural network.
Geometric deep learning has gained strong interest in recent years due to its ability to capture information in interpretable and generalizable representations; by using a group convolutional neural network, we inherit these characteristics for the context of color variation.
We represent color in the hue saturation luminance (HSL) color space, and leverage the insight that the hue, and saturation can be modeled with geometric group structure.
Using this approach, our network retains color information in a structured representation throughout the processing pipeline, thereby allowing color information to be used and/or discarded intentionally.

Recently, Color Equivariant Convolutional Networks [], proposed a group convolutional neural network architecture for hue equivariant representation learning.
Most notably, CEConv identifies hue transformations with the 2D rotation group. However, the proposed lifting layer introduces invalid RGB values which break equivariance, and degrade performance.
To resolve this issue, our lifting layer operates on the input image instead of the network filters, which produces an equivariant descriptor without projection induced artifacts.
This seemingly minor change improves equivariance error by more than three orders of magnitude, and stabilizes network performance across discretizations.
We also expand the notion of color equivariance to capture variation in saturation.
Specifically, we identify variations in saturation with the 1D translation group.
With this identification, we propose a group equivariant network where variations in hue and saturation are represented as a geometric transformations.

Figure: (a) Hue equivariance
Refer to caption: x1.png

## 2 Related Work

##### Group Convolutional Neural Networks

The strong generalization performance of convolutional neural networks on image processing tasks is attributed, in part, to the equivariance of planar convolution to 2D translations.
This insight has garnered considerable attention and led to a strong interest in the development of convolutional neural networks equivariant to other symmetry groups [; ].
Previous works introduce group convolutional networks for finite and continuous groups.
A framework for finite group convolution was developed in [], and demonstrated for the 2D rotation, and reflection groups.
The authors achieve equivariance to specific symmetry groups by convolving input images and feature maps with the group orbit of a learned filter bank.
In [], the authors design for equivariance to the continuous 2D rotation group by constraining filter representations to have circular harmonic structure.

Group convolutional networks have also been designed for use in the context of scale symmetry.
In [], the authors represent input images in log-polar coordinates where 2D rotation and scale transformations present as 2D translations.
To navigate the semi-group structure of scale transformations, the authors in [] approximate the scale space as finite and use dilated convolutions in a group equivariant architecture.
The SREN network proposed in [] designs for equivariance to the continuous 2D rotation and isotropic scaling group by constraining filter representations to be expressed as the linear combination of a windowed Fourier basis.

Group convolutional networks have also been designed for groups acting in higher dimensions.
For 3D rotational symmetry, [] and [] use a spherical harmonics based representation with 3D point clouds and spherical image inputs respectively; and [] introduce an equivariant network for transformations of the Euclidean group in 3D.
In [], the authors introduce a framework for general Lie groups whose exponential map is surjective, and [] introduces a framework for more general manifolds.
By designing for known symmetries in the task, these works provide improved interpretability, training efficiency and generalizability over conventional convolutional networks.

Other group equivariance methods encourage transformation equivariance using a soft penalty term on representation dissimilarity [; ].
These methods are separate from the group convolutional neural network literature, and different from what we propose.
Our model leverages the geometric structure of hue and saturation to design a hue-saturation equivariant network bringing the benefits of group equivariant networks to perceptual transformations.

##### Color Invariance

Several prior works attempt to mitigate the effect of color variation in image processing tasks, for example, in [], the authors represent images in a color space where pixel values are invariant to changes in luminance for luminance invariant image segmentation, and in [], the authors penalize changes in their latent representations due to color variation for color invariant skin lesion identification.
While color invariance resolves the effect of color variation, color has been identified as an important cue in neural network representation learning [; ] which is discordant with the goal of color invariance.
In contrast to these approaches, our work leverages the geometric structure of hue, and saturation to construct a convolutional neural network equivariant to variations in these quantities by design.

Most similar to ours is the model proposed in [], a hue equivariant network for improved network performance in the presence of color variation.
Our color equivariant networks differ from the work proposed in  [] in two important ways.
First, we lift the input image instead of the filters of the first layer which circumvents the issue of invalid hue rotations suffered by the network in [];
and second, we expand the notion of color equivariance by introducing networks that are equivariant to both hue and saturation.

## 3 Preliminaries

In this section we describe our notation, and review definitions of the group action, equivariance, and group convolution.

##### Notation.

We use $f^{l}$ to denote the $l$-th feature map ($f^{0}$ to denote an input image), and $f^{l}_{j}$ to denote the $j$-th channel of that feature map.
We use $\psi_{i}^{l}$ to denote the $i$-th filter of the $l$-th layer, and $\psi_{i,j}^{l}$ to denote the $j$-th channel of that filter.

##### Group action.

Adapted from [].
Given a set $X$ and a group $G$, the action of the group $G$ on $X$ is a function $\varphi:G\times X\rightarrow X$ satisfying the following:

- 1.
For all $g,h\in G$ and all $x\in X$
$\varphi(g,\varphi(h,x))=\varphi(gh,x)$
(1)
- 2.
For all $x\in X$
$\varphi(1,x)=x$
(2)
where $1\in G$ is the identity element of G.

The set $X$ is called a (left) $G$-set.

##### Equivariance.

Adapted from [].
Given two $G$-sets $X$ and $Y$, and group actions $\varphi:G\times X\rightarrow X$ and $\phi:G\times Y\rightarrow Y$, a function $f:X\rightarrow Y$ is said to be equivariant, if and only if for all $x\in X$, and $g\in G$,

$$ $f(\varphi(g,x))=\phi(g,f(x)).$ (3) $$

When $f$ is invariant to the action of $G$, the group action $\phi$ is the identity map and we write

$$ $f(\varphi(g,x))=f(x).$ (4) $$

The equivariance of our network to hue shifts is illustrated in Figure 1(a).

##### Group convolution.

In a conventional CNN, the input to convolution layer $l$, denoted $f^{l}:\mathbb{Z}^{2}\rightarrow\mathbb{R}^{K^{l}}$, is convolved with a set of $K^{l+1}$ filters, denoted $\psi^{l}_{i}:\mathbb{Z}^{2}\rightarrow\mathbb{R}^{K^{l}}$, where $i$ ranges from 1 to $K^{l+1}$.
The result of the convolution can be written:

$$ $f^{l+1}_{i}=[f^{l}*\psi_{i}^{l}](x)=\sum_{y\in\mathbb{Z}^{2}}\sum_{k=1}^{K^{l} }f^{l}_{k}(y)\psi^{l}_{i,k}(x-y),\;x\in\mathbb{Z}^{2}.$ (5) $$

The convolution of conventional CNNs is equivariant to the action of the group $(\mathbb{Z}^{2},+)$, that is, the group formed by summing over the integers.
The more general group convolution can be written:

$$ $[f^{l}*\psi_{i}^{l}](g)=\sum_{h\in G}\sum_{k=1}^{K^{l}}f^{l}_{k}(h)\psi_{i,k}^ {l}(h^{-1}g),$ (6) $$

and is equivariant to the action of the group $G$.

## 4 Method

In this section we present our color equivariant network.
We begin by presenting definitions for the hue and saturation groups and their group actions, then define the lifting layers, and group convolution layers for our color equivariant networks.

##### Hue group and group action.

In the HSL color space, hue is represented by angular position, and can therefore be identified with the 2D rotation group.
As in group convolutional network [], we consider a finite group representation.
Specifically, we identify elements of the discretized hue group, $H_{N}$, with those of the cyclic group $C_{N}$.

We define the action of hue group on HSL images, $x\in X$ where $x:\mathbb{Z}^{2}\rightarrow\mathbb{R}^{3}$, and functions on the discrete hue group, $y\in Y$, where $y:\mathbb{Z}^{2}\times H_{N}\rightarrow\mathbb{R}^{K}$.
An element of the hue group acts on an HSL image by the group action $\varphi_{h}:H_{N}\times X\rightarrow X$, which shifts the hue channel of the image.
Concretely, for an HSL image $x\in X$ defined as the concatenation of hue, saturation and luminance channels, i.e., $x=(x_{h},x_{s},x_{l})$, the action of an element $h_{i}$ of the hue group $H_{N}$ is given by

$$ $\varphi_{h}(h_{i},x)=((x_{h}+h_{i})(\text{mod}\;c),x_{s},x_{l}),$ (7) $$

where $(\cdot)\text{mod}(\cdot)$ denotes the modulus operation and is applied to the pixel value which ranges from 0 to $c$.

An element of the hue group acts on a function on the discrete hue group by the group action $\phi_{h}:H_{N}\times Y\rightarrow Y$, which “rotates" the function on the group.
Concretely, for a function $f$ on the discrete hue group $H_{N}$ defined as the concatenation of functions $f=(f_{1},\dots,f_{N})$, the action of an element $h_{i}$ in the hue group $H_{N}$ is given by

$$ $\phi_{h}(h_{i},f)=(f_{(1+i)(\text{mod}\;N)},\dots,f_{(N+i)(\text{mod}\;N)}).$ (8) $$

The action of $\varphi_{h}$ on an input image, and $\phi_{h}$ on a feature map are shown in Figure 6(a). We prove $\varphi_{h}$ and $\phi_{h}$ are group actions in Appendix A.

##### Saturation group and group action.

We expand the notion of color equivariance proposed in [] to include equivariance to saturation shifts.
In the HSL color space, saturation can be represented by a real number in the interval $[0,1]$; we introduce two approximations to give the saturation space group structure.
First, we observe that there is a bijection between the real numbers and the open interval $(0,1)$, so we can use the structure of the group $(\mathbb{R},+)$.
Second, we consider a finite subset of the group, a necessary and commonly used practice in both conventional CNNs (recall that the translation group is infinite) and group-equivariant CNNs such as []. With these approximations, we identify elements of the discretized saturation group $S_{N}$, with those of the integers with addition, $(\mathbb{Z},+)$.

We define the action of the saturation group on HSL images, $x\in X$ where $x:\mathbb{Z}^{2}\rightarrow\mathbb{R}^{3}$, and functions on the discrete saturation group, $y\in Y$, where $y:\mathbb{Z}^{2}\times S_{N}\rightarrow\mathbb{R}^{K}$.
An element of the saturation group acts on an HSL image by the group action $\varphi_{s}:S_{N}\times X\rightarrow X$, which shifts the saturation channel of the image.
Concretely, for an HSL image $x\in X$ defined as the concatenation of hue, saturation and luminance channels, i.e., $x=(x_{h},x_{s},x_{l})$, the action of an element $s_{i}$ of the saturation group $S_{N}$ is given by

$$ $\varphi_{s}(s_{i},x)=(x_{h},\text{min}(x_{s}+s_{i},c),x_{l}).$ (9) $$

An element of the saturation group acts on a function on the discrete saturation group by the group action $\phi_{s}:S_{N}\times Y\rightarrow Y$, which “translates" the function on the group.
Concretely, for a function $f$ on the discrete saturation group $S_{N}$ defined as the concatenation of functions $f=(f_{1},\dots,f_{N})$, the action of an element $s_{i}$ in the saturation group $S_{N}$ is given by

$$ $\phi_{s}(s_{i},f)=(f_{1+i},\dots,f_{N},\underbrace{\mathbf{0},\dots,\mathbf{0} }_{i}).$ (10) $$

The action of $\varphi_{s}$ on an input image, and $\phi_{s}$ on a feature map are shown in Figure 2. We prove $\varphi_{s}$ and $\phi_{s}$ are group actions in Appendix A.

Figure: Figure 2: Saturation equivariant feature maps. We illustrate the equivariance of our saturation-equivariant model. A saturation shift in the input image space (top-left to bottom-left), results in a feature map translation at each layer of the network (top-right to bottom-right). Corresponding feature maps are highlighted with a blue border.
Refer to caption: x6.png

##### Hue-Saturation group action.

We define the action of the hue-saturation group as a composition of the hue and saturation group actions.
An element of the hue-saturation group acts on an HSL image by the group action $\varphi_{hs}:H_{N}\times S_{M}\times X\rightarrow X$, which shifts both the hue and saturation channels of the image.
For an HSL image $x\in X$ defined as the concatenation of hue, saturation and luminance channels, i.e., $x=(x_{h},x_{s},x_{l})$, the action of an element $(h_{i},s_{j})$ of the hue-saturation group $H_{N}\times S_{M}$ is given by

$$ $\varphi_{hs}((h_{i},s_{j}),x)=\varphi_{h}(h_{i},\varphi_{s}(s_{j},x)).$ (11) $$

An element of the hue-saturation group acts on a function on the discrete saturation group by the group action $\phi_{hs}:H_{N}\times S_{M}\times Y\rightarrow Y$, which “rotates" and “translates" the function on the group.
Concretely, for a function $f$ on the discrete hue-saturation group $H_{N}\times S_{M}$ defined as the concatenation of functions $f=(f_{11},\dots,f_{1M},f_{21},\dots,f_{NM})$, the action of an element $(h_{i},s_{j})$ in the hue-saturation group $H_{N}\times S_{M}$ is given by

$$ $\phi_{hs}((h_{i},s_{j}),f)=\phi_{h}(h_{i},\phi_{s}(s_{j},f)).$ (12) $$

We prove $\varphi_{hs}$ and $\phi_{hs}$ are group actions in Appendix A.

##### Lifting layer.

The first layer of a group convolutional neural network “lifts" the input image to the group (see Figure 1(b)).
We can lift an input image to the product space of the image grid $\mathbb{Z}^{2}$, and discretized hue shifts $H_{N}=\{h_{0},h_{1},\dots,h_{N}\}$, by convolving with hue shifted filters,

$$ $[f^{0}*\psi^{0}_{i}](g_{x,j})=\sum_{y\in\mathbb{Z}^{2}}\sum_{k=1}^{K^{0}}f^{0} _{k}(y)h_{j}\psi^{0}_{i,k}(x-y).$ (13) $$

Figure: Figure 3: Impact of cardinality on hue rotation invertibility. Our lifting layer (blue) operates on HSL input images where each hue rotation is invertible. The lifting layer proposed in CEConv (orange) operates on RGB filters and suffers from invalid hue rotations for all discretizations of the hue group except for $N=1$ and $N=3$ (i.e., symmetries of the axis-aligned RGB cube). (Left) We show the impact of invalid hue rotations on a four pixel image. We rotate the image $60^{\circ}$ using our proposed lifting layer (top-left) and the CEConv lifting layer (bottom-left). Subsequently applying a $-60^{\circ}$ rotation yields an image that is indistinguishable from the original using our approach (top-left), and one with visible artifacts using the CEConv approach (bottom-left). (Right) We show the average restored image error for both approaches. Our approach results in a consistently negligible restored image error, however, the CEConv approach results in a restored image error exceeding $7\%$ for all discretizations of the hue group except $N=1$ and $N=3$.
Refer to caption: x7.png

Here we denote an element of the product space $g_{x,i}$, where the subscript $x$ references an element of $\mathbb{Z}^{2}$, and $j$ references the element $h_{j}$ in $H_{N}$.
The input image $f^{0}$ and lifting filters $\psi_{m,n}$ are functions on $\mathbb{Z}^{2}$.
This is the approach taken in CEonv []. The authors shift the hue of a filter in the RGB space by rotating its values about an axis passing through the point $p=(1,1,1)$ as shown in Figure 8 in Appendix B.
As described in [], this approach results in invalid hue rotations for all discretizations of the hue group that are not symmetries of the axis-aligned RGB cube (i.e., $N=1$ and $N=3$).
To remedy this, the authors project invalid RGB values to the nearest point on the RGB cube.
However, this projection breaks invertibility of the hue shift (see Figure 3) yielding a descriptor that is only approximately hue equivariant.

Lifting to the group can alternatively be achieved by transforming the input image rather than the filters,

$$ $f^{1}(g_{x,j})=\varphi_{h}(h_{j},f^{0})(x).$ (14) $$

We use this approach in our implementation as it circumvents the issue of invalid hue rotations, producing a reliably equivariant descriptor.
The lifting layers for the saturation group, and hue-saturation group are constructed analogously.

##### Equivariance of the lifting layer.

We can show that our lifting layer is equivariant to hue shifts.
We let $f^{1}(g_{x,j})=\varphi_{h}(h_{j},f^{0})(x)$ as in equation 14 and show that a hue shift of $f^{0}$ results in a hue shift of $f^{1}$.
In the first step we use the fact that $\varphi_{h}$ is a group action (see Appendix A.1), and in the second step we use commutativity of the hue group:

$$ $\displaystyle\varphi_{h}(h_{j},\varphi_{h}(h_{m},f^{0}))(x)$ $\displaystyle=\varphi_{h}(h_{j}h_{m},f^{0})(x)$ (15) $\displaystyle=\varphi_{h}(h_{m}h_{j},f^{0})(x)$ (16) $\displaystyle=\varphi_{h}(h_{m},\varphi_{h}(h_{j},f^{0}))(x)$ (17) $\displaystyle=\varphi_{h}(h_{m},f^{1}(g_{x,j})).$ (18) $$

Equivariance of the saturation, and hue-saturation lifting layers can be shown analogously.

##### Group convolution layer.

For all layers after the first layer, the feature maps $f^{l}$ are on the product space $\mathbb{Z}^{2}\times H_{N}$.
Since hue shifts can only be performed on three dimensional inputs, we leverage the identification of hue shifts with the discrete 2D rotation group, i.e., $H_{N}\cong C_{N}$ and interpret the feature maps $f^{l}$ as functions on the product space $G:=\mathbb{Z}^{2}\times C_{N}$.
With this interpretation we

Figure: Figure 4: Model sample efficiency. We show the error improvement (higher is better) over the Z2CNN baseline as a function of the percentage of training examples used. The advantage of our Hue-$N$ models increases as the percentage of training examples used decreases.

Table: Table 1: Generalization to global hue-shift. Classification error on the Hue-shift MNIST dataset is reported. Our hue-equivariant models (Hue-$N$) perform comparably to the conventional CNN model (Z2CNN) on the in-distribution case ($A/A$), and significantly outperform the conventional CNN model on the out-of-distribution case ($A/B$).

| Network | $A/A$ | $A/B$ | Params |
| --- | --- | --- | --- |
| Z2CNN | 1.54 (0.10) | 57.38 (22.06) | 22,130 |
| Hue-3* | 1.79 (0.25) | 1.81 (0.29) | 22,658 |
| Hue-4* | 1.97 (0.25) | 2.08 (0.15) | 25,690 |

perform group convolution on $G$ as follows,

$$ $[f^{l}*\psi^{l}_{i}](g)=\sum_{h\in H}\sum_{k=1}^{K^{0}}f^{l}_{k}(h)\psi^{l}_{i ,k}(g^{-1}h),$ (19) $$

where $\psi^{l}$ is a function on $G$. Group convolution on the saturation, and hue-saturation groups are performed analogously.

## 5 Experiments

In this section, we highlight improved sample efficiency over a conventional CNN baseline, improved hue-equivariance over CEConv [], the utility of saturation equivariance on the Camelyon17 dataset [], and the utility of our architectures for color based sorting.
We additionally demonstrate the strong performance of our model through extensive experiments and against numerous baselines.

### 5.1 Hue-shift MNIST Classification

We demonstrate improved generalization to global hue-shifts, and higher sample efficiency compared to a conventional CNN model with a similar number of parameters on our Hue-shift MNIST dataset.
Our dataset is a variation of MNIST [] with 60k training examples and 10k test examples classified into one of 10 categories.
Additional details are provided in Appendix C.1.

Figure: (a) Z2CNN
Refer to caption: x9.png

##### Generalization to global hue-shift.

The generalization performance of our hue-equivariant models, Hue-$N$ ($N$ indicates the cardinality of the discrete hue group), and a conventional CNN model, Z2CNN [], are reported in Table 4.
On the in-distribution test case ($A$/$A$), the performance of our model and the conventional CNN model are comparable.
However, on the out-of-distribution test case ($A$/$B$), the performance of our model is preserved, while the performance of the conventional CNN model deteriorates significantly.

The performance gap on the out-of-distribution test case can be attributed to the difference in internal representation structure.
To understand this better, we generate feature representation trajectories in the penultimate layer of our network and the baseline architecture by continuously varying the hue of an input image. We then visualize these trajectories using tSNE [] projection (see Figure 5).
For the baseline architecture, the trajectories of different digits overlap for some hues (Figure 5(a)), but for our model, the trajectories of different digits are confined to separate clusters (Figure 5(b)).

##### Model sample efficiency.

The sample efficiency of our hue-equivariant models (Hue-$N$) and a conventional CNN model (Z2CNN) are reported in Figure 4.
The advantage of our hue-equivariant models increases as the percentage of the training set used decreases, illustrating improved sample efficiency [].

Figure: (a) Hue-shifted feature maps
Refer to caption: x11.png

### 5.2 Hue-shift 3D Shapes Classification

We demonstrate improved generalization to local hue-shifts, and significantly reduced equivariance error
compared to CEConv on the 3D Shapes classification dataset [].
The 3D Shapes dataset consists of RGB images of 3D shapes, where the color of the shape, the floor, and the walls vary across examples, as does the scale and orientation of the shape.
There are 48k examples in the training set, and 12k examples in each of the test sets. Additional details are provided in Appendix C.2.

**Table 2: Generalization to local hue-shift. Classification error on the 3D Shapes dataset is reported. Our model (Hue-$N$) and CEConv-$N$ achieve improved generalization performance over the conventional CNN model (Z2CNN). Our approach is robust to the choice of hue group discretization, while the performance of CEConv deteriorates when $N$ is not a symmetry of the axis-aligned RGB cube.**
| Network | $A$/$A$ | $A$/$B$ | $A$/$C$ | Params |
| --- | --- | --- | --- | --- |
| Z2CNN | 0.00 (0.00) | 51.25 (9.59) | 26.66 (19.60) | 20,192 |
| Hue-3* | 0.00 (0.00) | 0.03 (0.04) | 0.04 (0.03) | 19,478 |
| Hue-4* | 0.00 (0.00) | 0.00 (0.00) | 0.00 (0.00) | 21,832 |
| CEConv-3 | 0.00 (0.00) | 0.02 (0.03) | 0.05 (0.04) | 26,441 |
| CEConv-4 | 0.00 (0.03) | 0.60 (0.25) | 0.53 (0.43) | 30,792 |

##### Generalization to local hue-shift.

The generalization performance of our hue-equivariant models, Hue-$N$ ($N$ indicates the cardinality of the discrete hue group), a conventional CNN model, Z2CNN [], and the CEConv-$N$ models are reported in Table 2.
Our Hue-4 model outperforms all other models on the in-distribution ($A/A$), global hue-shift out-of-distribution ($A/B$) and local hue-shift out-of-distribution ($A/C$) test sets.
The performance of CEConv-4 is significantly worse than all other hue-equivariant models.
We attribute the relatively poor generalization performance of the CEConv-4 model to the feature map equivariance error.

##### Lifting layer equivariance error.

The lifting layer equivariance error of our Hue-4 model and the CEConv-4 model are reported in Figure 6(b).
As discussed in section 4, our lifting layer operates on HSL images and does not induce the invalid hue rotations observed in CEConv.
We quantitatively assess how lifting layer invertibility impacts descriptor equivariance following [].
We compute the equivariance error as the relative equivariance,

$$ $\text{Equivariance Error}=\frac{|f(\varphi_{h}(h_{i},x))-\phi_{h}(h_{i},f(x))| }{|f(\varphi_{h}(h_{i},x))+\phi_{h}(h_{i},f(x))|}$ (20) $$

and show that the equivariance error of CEConv-4 is more than three orders of magnitude higher than our Hue-4 model (see Figure 6(b)).

### 5.3 Camelyon17 Classification: Saturation shift in the wild

We demonstrate improved generalization to saturation-shifts compared to ResNet50 [] and CEConv-3 on the Camelyon17 classification dataset [].
The Camelyon17 dataset consist of images of human tissue collected from five different hospitals.
Variation in tissue images results from variation in the data collection and processing procedures.
The dataset consists of 387,490 examples, 302,436 of which are used for training and 85,054 of which are used for testing.
Hospitals 1, 2, and 3 are represented in the training set, and hospital 5 is represented in the testing set.
Additional details are provided in Appendix D.3.

**Table 3: Generalization to saturation-shift. Classification error on the Camelyon17 dataset is reported. Our model (Sat-$N$ and Hue-$M$-Sat-$N$) achieves improved generalization performance over the conventional CNN model (ResNet50).**
| Networks | Error | Param |
| --- | --- | --- |
| ResNet50 | 28.91 (7.58) | 23.5M |
| Sat-3* | 16.08 (2.68) | 23.3M |
| Hue-4-Sat-3* | 19.06 (4.92) | 23.0M |
| CEConv-3 | 28.76 (9.93) | 23.1M |

##### Generalization to saturation-shift.

We expand the notion of color equivariance proposed in [] to capture variation in saturation.
The generalization performance of our saturation-equivariant models, Resnet50 [], and CEConv-3 are reported in Table 3.
Our saturation- and color-equivariant models outperforms all other models, while CEConv performs comparably to the conventional CNN model.

### 5.4 Color shift in the wild

We demonstrate improved generalization to global color-shifts compared to ResNet [] and CEConv on the Caltech-101 [], Oxford-IIT Pets [], Stanford Cars [], STL-10 [], CIFAR-10 and CIFAR-100 []. Additional details are provided in Appendix D.

**Table 4: Generalization to color-shift. Classification error on the Caltech-101, Oxford-IIT Pets, Stanford Cars, CIFAR-10, CIFAR-100, STL-10 datasets are reported. Our models (Hue-$N$, Sat-$N$ and Hue-$M$-Sat-$N$) outperform CEConv-$N$ models on all datasets, and are competitive with ResNet.**
| Network | Caltech 101 | CIFAR-10 | CIFAR-100 | Stanford Cars | Oxford Pets | STL-10 |
| --- | --- | --- | --- | --- | --- | --- |
| ResNet | 32.68 (1.55) | 7.86 (1.14) | 32.00 (0.63) | 25.41 (0.96) | 31.52 (2.05) | 18.59 (1.65) |
| Hue-3* | 34.32 (0.34) | 8.41 (0.39) | 33.28 (0.63) | 22.19 (2.20) | 31.37 (3.50) | 19.82 (0.98) |
| Hue-4* | 32.23 (1.07) | 8.83 (0.64) | 34.70 (0.89) | 20.38 (1.06) | 27.39 (0.68) | 20.73 (1.11) |
| Hue-4-Sat-3* | 38.14 (1.07) | 10.68 (0.78) | 33.27 (0.31) | 24.79 (3.87) | 29.84 (1.34) | 20.53 (0.73) |
| Sat-3* | 41.64 (1.40) | 9.24 (0.27) | 39.33 (0.45) | 31.90 (10.03) | 36.87 (5.57) | 20.71 (1.10) |
| CEConv-3 | 34.74 (0.83) | 8.86 (0.33) | 34.95 (0.44) | 23.97 (1.56) | 31.08 (2.54) | 24.29 (1.31) |
| CEConv-4 | 33.52 (0.48) | 9.28 (0.24) | 35.46 (0.35) | 24.08 (0.66) | 33.70 (1.50) | 21.90 (1.64) |

##### Generalization to color-shift.

The generalization performance of our hue-equivariant models, Hue-$N$, saturation-equivariant model, Sat-$N$, color-equivariant models Hue-$M$-Sat-$N$, ResNet44 model, and the CEConv-$N$ models are reported in Table 4.

##### Color sorting on CIFAR-10.

While our hue-equivariant model performs on-par with the conventional CNN model on this dataset, the structure of our representation can be leveraged for tasks that are not possible with conventional architectures alone.
To illustrate this, we use our model to sort images in the automobile class by hue (see Figure 7(a)).
To produce an hue ordering, we compute the hue-shifted pairwise Euclidean distance between hue group feature maps of instances in the automobile class.
Images $x_{1}$ and $x_{2}$ are close in the hue space if

$$ $\arg\min_{i}\|\phi_{h}(h_{i},\Psi(x_{1})),\Psi(x_{2})\|=0,$ (21) $$

where $\Psi(x_{j})$ denotes the penultimate feature map for image $j$ and $h_{i}$ is the $i$-th element of the hue group.
Sorting using the penultimate feature map of the ResNet44 model is shown in Figure 7(b).

Figure: (a) Hue-$4$
Refer to caption: x12.png

## 6 Conclusion

In this paper we address the challenge of learning hue- and saturation-equivariant representations.
Leveraging the observation that these perceptual transformations have geometric structure, we propose a group structure for each, and a group convolutional neural network that is equivariant to the transformations by design.
By encoding perceptual variation in a finite group structure, we achieve strong generalization performance and sample efficiency and can use learned representations for tasks such as color based sorting.

### 6.1 Limitations and future work

While our method is equivariant to hue and saturation by design, it is not guaranteed to be equivariant to luminance.
However, since luminance is computed as a linear combination of the red, green and blue channel values of an input image, if the preprocessing statistics (e.g., data mean) on the training set are valid for the testing set, luminance is approximately unchanged.
Additional limitations come from our choice of equivariant architecture.
Group convolutional neural networks are more computationally expensive than their conventional counterparts since they require computation of the filter orbit at each layer.
This is compounded with the requirement for a finite filter orbit which can at best approximate a continuous group.
Future work will investigate a continuous representation of the group to address both of the aforementioned challenges.

## 7 Reproducibility Statement

Source code will be open-sourced upon acceptance.
Proofs for hue and saturation group actions are shown in Appendix A.
Proof of equivariance of the proposed lifting layer is shown in Section 4.
Datasets, training and implementation details are described in Appendix C.

## Appendix

### A Hue and saturation group action

In this section we prove the proposed hue group action and saturation group action satisfy the axioms for a group action. By definition of a group action, $\varphi_{h}$ is a group action if it satisfies the following properties:

- 1.
For all $g,h\in G$ and all $x\in X$
$\varphi(g,\varphi(h,x))=\varphi(gh,x)$
(22)
- 2.
For all $x\in X$
$\varphi(1,x)=x$
(23)
where $1\in G$ is the identity element of G.

In the remainder of this section we refer to equation 22 as ‘property one’ and equation 23 as ‘property two’.

#### A.1 Hue group action

Here we show that the proposed hue group action is indeed a group action. The proposed hue group action on the input space is defined:

$$ $\varphi_{h}(h_{i},x)=((x_{h}+h_{i})(\text{mod}\;255),x_{s},x_{l}),$ (24) $$

in equation 7 of the main text.

First we show the proposed hue group action $\varphi_{h}$ satisfies property one. For any hue shifts $h_{i}$ and $h_{j}$, we have

$$ $\displaystyle\varphi_{h}(h_{i},\varphi_{h}(h_{j},x))$ $\displaystyle=\varphi_{h}(h_{i},((x_{h}+h_{j})(\text{mod}\;255),x_{s},x_{l}))$ (25) $\displaystyle=(((x_{h}+h_{j})(\text{mod}\;255)+h_{i})(\text{mod}\;255),x_{s},x _{l}))$ (26) $\displaystyle=(((x_{h}+(h_{j}+h_{i}))(\text{mod}\;255),x_{s},x_{l}))$ (27) $\displaystyle=\varphi_{h}((h_{j}+h_{i}),x),$ (28) $$

which shows that $\varphi_{h}$ satisfies the property one.
Now we show $\varphi_{h}$ satisfies property two. We have the identity element hue transformation $h_{0}=0$, and

$$ $\displaystyle\varphi_{h}(h_{0},x)$ $\displaystyle=((x_{h}+h_{0})(\text{mod}\;255),x_{s},x_{l})$ (29) $\displaystyle=(x_{h},x_{s},x_{l})=x,$ (30) $$

which shows $\varphi_{h}$ satisfies property two.
Having shown that $\varphi_{h}$ satisfies all properties of a group action, we have that $\varphi_{h}$ is, in fact, a group action.

#### A.2 Saturation group action

Here we show the proposed saturation group action is a group action.
The proposed saturation group action is defined:

$$ $\varphi_{s}(s_{i},x)=(x_{h},\text{min}(x_{s}+s_{i},255),x_{l}),$ (31) $$

in equation 9 of the main text.
First we show the proposed saturation group action $\varphi_{s}$ satisfies property one.
For any saturation shifts $s_{i}$ and $s_{j}$, we have

$$ $\displaystyle\varphi_{s}(s_{i},\varphi_{s}(s_{j},x))$ $\displaystyle=\varphi_{s}(s_{i},(x_{h},\text{min}(x_{s}+s_{j},255),x_{l}))$ (32) $\displaystyle=(x_{h},\text{min}(\text{min}(x_{s}+s_{j},255)+s_{i},255),x_{l})$ (33) $\displaystyle=(x_{h},\text{min}(x_{s}+(s_{j}+s_{i}),255),x_{l})$ (34) $\displaystyle=\varphi_{s}((s_{j}+s_{i}),x),$ (35) $$

which shows $\varphi_{s}$ satisfies property one.
Now we show $\varphi_{s}$ satisfies property two.
We have the identity element saturation transformation $s_{0}=0$, and

$$ $\displaystyle\varphi_{s}(s_{0},x)$ $\displaystyle=(x_{h},\text{min}(x_{s}+s_{0},255),x_{l})$ (36) $\displaystyle=(x_{h},x_{s},x_{l})=x,$ (37) $$

which shows $\varphi_{s}$ satisfies property two.
Having shown that $\varphi_{s}$ satisfies all properties of a group action, we have that $\varphi_{s}$ is, in fact, a group action.

### B Invalid hue rotation in CEConv

Figure: Figure 8: CEConv lifting. CEConv shift the hue of a filter in the RGB space by rotating its values about an axis passing through the point $p=(1,1,1)$. This approach results in invalid hue rotations for all discretizations of the hue group that are not symmetries of the axis-aligned RGB cube (i.e., $N=1$ and $N=3$). (Top) $N=3$. Rotations symmetric to axis-aligned RGB cube and yields valid rotations. (Bottom) $N=4$. Rotations yield results where invalid RGB values that lie outside of the RGB cube and needs to be projected.
Refer to caption: extracted/5941270/figures/CEN_ICLR_RGBCubeRotation.png

### C Datasets

#### C.1 Hue-shift MNIST

We introduce the Hue-shift MNIST dataset to evaluate the performance of our equivariant architectures in the presence of global hue shift.
Examples in the training set are randomly assigned a hue between $0^{\circ}$ and $240^{\circ}$ (see Figure 9(a)) and examples in the test set are partitioned into an in-distribution set and an out-of-distribution set.
In-distribution examples are randomly assigned a hue from the same distribution as the training set, and out-of-distribution examples are randomly assigned a hue outside of the training set distribution (i.e., between $240^{\circ}$ and $360^{\circ}$) (see Figure 9(b)).

Figure: (a) Dataset $A$ examples
Refer to caption: extracted/5941270/figures/mnist_train.png

Figure: (a) Dataset $A$ examples
Refer to caption: extracted/5941270/figures/shapes3D_hue_train.png

#### C.2 Hue-shift 3D Shapes

We introduce the Hue-shift 3D Shapes dataset to evaluate the performance of our equivariant architectures in the presence of local hue shift.
Examples in the training set are randomly assigned a hue from the first half of the color space (colors 0-4 as defined in Burgess and Kim []) and examples in the test set are are partitioned into an in-distribution set and two out-of-distribution sets.
The out-of-distribution test sets are designed to measure robustness to global hue-shift, and local hue-shift.
In the out-of-distribution test set designed to measure robustness to global hue-shift, the color of the wall, the floor and the shape are randomly selected from the second half of the color space (colors 5-9 as defined in Burgess and Kim []), and in the out-of-distribution test set designed to measure robustness to local hue-shift, the color of the wall and the floor are randomly selected from the first half of the color space, and the color of the shape is randomly selected from the second half of the color space.

### D Implementation and training details

In this section we provide architectural details for our equivariant networks, and training details for each experiment.

All experiments were performed over multiple random seeds to assess the robustness of the model to initialization. Performance statistics on the Hue-shift MNIST, Hue-shift 3D Shapes, and CIFAR-10 datasets were computed over three random seeds (i.e., 1999, 2000, and 2001). Performance statistics the on Camelyon17 dataset were computed over five random seeds (i.e., 1997, 1998, 1999, 2000, and 2001).

All models were trained on a shared research computing cluster. Each compute node allocates an Nvidia L40 GPU, 24 core partitions of an Intel Xeon Gold 5320 CPU, and 24GBs of DDR4 3200MHz RDIMMs.

#### D.1 Hue-shift MNIST

For this task, we compare against the Z2CNN architecture proposed in Cohen and Welling [].
Our hue-equivariant architecture has the same number of layers as Z2CNN, but with a reduced filter count to maintain a similar number of parameters at each layer.
In the final layer of our hue-equivariant architecture, we perform hue group pooling to yield an hue-invariant representation.

We train our equivariant networks and the conventional architectures for 5 epochs with a batch size of 128.
We optimize over a cross-entropy loss using the Adam optimizer Kingma and Ba [] with $\beta_{1}=0.9$, and $\beta_{2}=0.999$.
We use an initial learning rate of $10^{-3}$ for the $\mathbb{Z}^{2}$ network, and $10^{-4}$ for the hue-equivariant network.
We train CEConv architectures using the hyperparameters reported in .

#### D.2 Hue-shift 3D Shapes

For this task we compare against the Z2CNN architectures proposed in Cohen and Welling [].
Our hue-equivariant architectures are designed with the same network structure as the baselines, but with a reduced filter count to maintain a similar number of parameters at each layer.
In the final layer of our hue-equivariant architectures, we perform hue group pooling to yield an hue-invariant representation.

Following Wiles et al. [], we train the ResNet architectures for 100k iterations with a batch size of 128.
We optimize over a cross-entropy loss using SGD with a learning rate of $10^{-2}$.
Images are down-sampled by a factor of 2 to train the CNN architectures. CNN architectures are trained as described in Section D.1.

#### D.3 Camelyon17

For this task, we compare against the ResNet50 architecture He et al. [].
Our hue- and saturation-equivariant architectures have the same number of layers as ResNet50, but with a reduced filter count to maintain a similar number of parameters at each layer.
In the final layer of our hue- and saturation-equivariant architectures, we perform a group pooling to yield group invariant representation.

We train our equivariant networks and the conventional architectures for 10k iterations with a batch size of 32.
We optimize over a cross-entropy loss using the Adam optimizer Kingma and Ba [] with an initial learning rate of $10^{-2}$, $\beta_{1}=0.9$, and $\beta_{2}=0.999$.
We train CEConv architectures using the hyperparameters reported in .

Figure: Figure 11: Camelyon-17 dataset saturation statistics. Example images and saturation statistics by node (hospital).
Refer to caption: extracted/5941270/figures/CEN_CamelyonDistribution.png.png

We present experimental results for four discretizations of the saturation space. Our choice of discretizations, i.e., $d\in\{\nicefrac{{1}}{{20}},\nicefrac{{1}}{{10}},\nicefrac{{3}}{{20}},\nicefrac
{{1}}{{2}}\}$, are determined with consideration of the saturation range of the training and validation set (see Figure 11).
For a given choice of discretization $d$, an element $s_{i}\in S_{N}$ shifts the saturation of an input image by $s_{i}=i*d*c$, where $c$ is the maximum pixel value.
We limit computational expense of lifting and convolution by performing these operations over the truncated set $\{s_{-1},s_{0},s_{1}\}$ for our Sat-3 model.

#### D.4 CIFAR-10

For this task, we compare against the ResNet44 architecture proposed in Cohen and Welling [].
Our hue-, saturation- and color-equivariant architectures have the same number of layers as ResNet44, but with a reduced filter count to maintain a similar number of parameters at each layer.
In the final layer of our networks, we perform group pooling to yield group invariant representation.

We train our equivariant networks and the conventional architectures for 300 epochs, and a batch size of 128.
We optimize over a cross-entropy loss using SGD with an initial learning rate of $10^{-1}$ and a cosine-annealing scheduler.
We evaluate the performance of our model in the presence of natural hue shifts on the CIFAR-100 Krizhevsky et al. [] classification dataset.
The dataset consists of 60k examples, 50k of which are used for training and 10k for testing.

#### D.5 CIFAR-100

For this task, we compare against the ResNet44 architecture proposed in Cohen and Welling [].
Our hue-, saturation- and color-equivariant architectures have the same number of layers as ResNet44, but with a reduced filter count to maintain a similar number of parameters at each layer.
In the final layer of our networks, we perform group pooling to yield group invariant representation.

We train our equivariant networks and the conventional architectures for 300 epochs, and a batch size of 128.
We optimize over a cross-entropy loss using SGD with an initial learning rate of $10^{-1}$ and a cosine-annealing scheduler.
We evaluate the performance of our model in the presence of natural hue shifts on the CIFAR-100 Krizhevsky et al. [] classification dataset.
The dataset consists of 60k examples, 40k of which are used for training, 10k for validation, and 10k for testing.

#### D.6 Caltech-101

For this task, we compare against the ResNet18 architecture proposed in He et al. [].
Our hue-, saturation- and color-equivariant architectures have the same number of layers as ResNet18, but with a reduced filter count to maintain a similar number of parameters at each layer.
In the final layer of our networks, we perform group pooling to yield group invariant representation.

We train our equivariant networks and the conventional architectures for 300 epochs, and a batch size of 16.
We optimize over a cross-entropy loss using Adam Kingma and Ba [] with an initial learning rate of $10^{-2}$.
We evaluate the performance of our model in the presence of natural hue shifts on the Caltech-101 Li et al. [] classification dataset.
The dataset consists of 9,146 examples, $\nicefrac{{2}}{{3}}$ of which are used for training and $\nicefrac{{1}}{{3}}$ for testing.

#### D.7 STL-101

For this task, we compare against the ResNet18 architecture proposed in He et al. [].
Our hue-, saturation- and color-equivariant architectures have the same number of layers as ResNet18, but with a reduced filter count to maintain a similar number of parameters at each layer.
In the final layer of our networks, we perform group pooling to yield group invariant representation.

We train our equivariant networks and the conventional architectures for 300 epochs, and a batch size of 16.
We optimize over a cross-entropy loss using Adam Kingma and Ba [] with an initial learning rate of $10^{-2}$.
We evaluate the performance of our model in the presence of natural hue shifts on the STL-10 Coates et al. [] classification dataset.
The dataset consists of 5k training examples and 8k testing examples.

#### D.8 Stanford Cars

For this task, we compare against the ResNet18 architecture proposed in He et al. [].
Our hue-, saturation- and color-equivariant architectures have the same number of layers as ResNet18, but with a reduced filter count to maintain a similar number of parameters at each layer.
In the final layer of our networks, we perform group pooling to yield group invariant representation.

We train our equivariant networks and the conventional architectures for 300 epochs, and a batch size of 16.
We optimize over a cross-entropy loss using Adam Kingma and Ba [] with an initial learning rate of $10^{-2}$.
We evaluate the performance of our model in the presence of natural hue shifts on the Stanford Cars Krause et al. [] classification dataset.
As the original host for the Stanford cars dataset is no longer maintained, the dataset was constructed from the images Kaggle [], the devkit github [], and the annotation files Kaggle [].
The dataset consists of 8,144 training examples and 8,041 testing examples.

#### D.9 Oxford-IIT Pets

For this task, we compare against the ResNet18 architecture proposed in He et al. [].
Our hue-, saturation- and color-equivariant architectures have the same number of layers as ResNet18, but with a reduced filter count to maintain a similar number of parameters at each layer.
In the final layer of our networks, we perform group pooling to yield group invariant representation.

We train our equivariant networks and the conventional architectures for 300 epochs, and a batch size of 16.
We optimize over a cross-entropy loss using Adam Kingma and Ba [] with an initial learning rate of $10^{-2}$.
We evaluate the performance of our model in the presence of natural hue shifts on the Oxford-IIT Pets   classification dataset.
The dataset consists of 3,680 training examples and 3,669 testing examples.

## Peer Reviews

confidence: 2
contribution: 2
initial_rating: 5
presentation: 3
questions: Would the proposed method improve image classification performance on large-scale datasets like ImageNet or on tasks beyond image classification?
soundness: 2
strengths: 1.	The approach is technically compelling, as it enables convolutional neural networks to be equivariant to hue and saturation by design, leveraging the inherent geometric structure of these color components.

2.	Results across several benchmarks indicate that the proposed method offers stronger robustness to color variation compared to existing methods.

3.	The paper is well-structured and easy to understand.
summary: This paper introduces group convolutional neural networks designed to be equivariant to variations in hue, saturation, and color in RGB images. It proposes a lifting layer that directly transforms the input image to produce a color-equivariant descriptor, effectively addressing the challenge of invalid RGB values found in previous works. Experiments on synthetic and real-world datasets highlight the robustness of the proposed method to color variations.
weaknesses: 1.	Experiments are limited to small toy datasets with pronounced hue or saturation shifts, leaving it unclear how the method would perform on large datasets like ImageNet.

2.	The proposed method focuses on hue and saturation equivariance, but does not address other common variations in real-world data, such as 3D geometric transformations or lighting changes.

---

confidence: 4
contribution: 3
initial_rating: 8
presentation: 3
questions: See weakness part.
soundness: 4
strengths: 1. The paper is very well written and easy to read.

2. The authors conducted extensive experiments using natural images (CIFAR10, CIFAR100, STL-10, etc.) as well as medical images (Camelyon-17) to validate the effectiveness of their algorithm. The experiments are thorough and detailed.

3. Color equivariant representations are an important concept in computer vision, yet they are often overlooked in contemporary research. The authors effectively claim the significance of color in the era of deep learning in this work.
summary: The paper extends the concept of color equivariance to include saturation shifts, in addition to hue transformations, then they design GCNNs that are equivariant to color variation by leveraging the geometric structure of color in the hue-saturation-luminance (HSL) color space, additionally a novel lifting layer is proposed to transform input images directly, avoiding invalid RGB values and significantly improving equivariance error.
weaknesses: I believe this work has no significant drawbacks and meets the acceptance standards for ICLR. 

Beyond that, I am curious about two questions:

1. This paper uses the HSL color space as the basis for designing experiments, specifically designing groups and group actions for hue and saturation, as well as a hue-saturation group action. I am interested to know whether this approach would also be applicable to other chromacity-luminance color spaces, such as CIELAB or YUV.  Additionally, the experiments are designed around hue shifts in the HSI series (i.e. H in HSV. HSL, HSB...). Could other chromacity shifts in different color spaces be designed to further demonstrate the robustness of the model?

2. The experiments presented in this paper focus on classification tasks. I wonder if this design could be even more beneficial for higher-level tasks or those that rely more on semantic information, such as semantic segmentation. Do the authors have any plans to design similar experiments for such tasks?

---

confidence: 3
contribution: 2
initial_rating: 6
presentation: 3
questions: 1, The additional computational load from group convolutions and lifting layers present a challenge for real-time applications. Therefore, it is essential to compare your method with common alternatives, such as color augmentation or robust contrastive learning approaches, which are known to enhance robustness to color and perceptual variations. These comparisons would be especially valuable for demonstrating the unique advantages of color equivariance over standard augmentation techniques.
soundness: 3
strengths: 1, The paper presents a novel approach to color-equivariant convolutional neural networks (CNNs) by extending GCNNs to handle hue and saturation changes in the HSL color space. This is achieved with an inventive lifting layer that acts directly on input images, improving stability and reducing errors by avoiding invalid RGB values—a common issue in previous architectures.

2, The authors demonstrate a thorough experimental design using various synthetic and real-world datasets (e.g., Hue-shift MNIST, Camelyon17) known for color variations, showing the model’s robustness. Both quantitative and qualitative analyses, including sample efficiency and feature map visualizations, offer a detailed evaluation. Comparisons with competitive baselines like CEConv and ResNet confirm the model’s generalizability and efficiency.

3, The paper is well-structured, with a logical flow from problem statement to methodology and results. Clear explanations of the hue- and saturation-equivariant group actions, along with detailed figures, aid in understanding the technical concepts. Comprehensive appendices further enhance reproducibility and transparency by providing proofs and additional experiment details.
summary: The paper presents a novel convolutional neural network (CNN) architecture tailored to handle color variations effectively. The authors introduce a color-equivariant group convolutional network (GCNN) that is robust to changes in hue and saturation, significantly improving performance in tasks involving perceptual color transformations. By employing a new lifting layer that operates directly on input images rather than filters, the model overcomes limitations of existing color-equivariant networks, achieving enhanced stability and reducing equivariance errors by orders of magnitude. This approach yields superior performance on both synthetic and real-world datasets, particularly excelling in generalizing to out-of-distribution color variations and enhancing sample efficiency.
weaknesses: 1, The paper extends GCNNs to be equivariant to hue and saturation but does not address luminance variations, which are also important for color representation. Although the authors suggest that luminance invariance may be approximately preserved, this may not hold in real-world applications, such as medical imaging or scenes with varying lighting. Future work could integrate luminance into equivariant transformations or evaluate its impact on specific tasks.

2, The model is compared to standard CNNs (e.g., ResNet) and color-equivariant networks (e.g., CEConv), but not to other common approaches for handling color variations, such as color augmentation techniques or robust contrastive learning models. Adding these comparisons would strengthen the case for this method’s effectiveness in handling color shifts.

---

confidence: 3
contribution: 2
initial_rating: 5
presentation: 3
questions: 1/ Why not use the experimental protocol of [16] that seems well detailed in their paper for Table 4?

2/ What would be the pros and cons of using the proposed approach vs the SOTA (or a strong baseline from the leaderboard such as Group DRO) on Camelyon17 ?

3/ Could you explain how the different cardinalities have been chosen for the different experiments?

4/ In the limitation section, the computation overhead of the approach is mentioned, without mentioning explicit numbers compared to the baseline’s training time, could this information be mentioned?
soundness: 4
strengths: * The paper is well written and illustrated.
* The approach is well motivated and addresses concrete tasks, for instance improving classification robustness in case of different imaging processes coming from different labs in the case of medical imagery. 
* The claims are supported by experiments. 
* Some limitations of the approach are presented.  
* The code will be open source.
* The appendix provides necessary information to reproduce results.
* The approach designs color equivariance in group CCNN models, which is an interesting property.
summary: This paper builds on the approach “Color Equivariant Convolutional Networks” [16, Neurips 2023], modifying their lifting layer by operating on the input image rather than on the filters. Additionally, it generalizes the equivariance, previously only defined on the hue channel to the  saturation channel. Experiments on synthetic datasets (hue shift MNIST and hue shift 3D shape) and on CIFAR, Caltech, Camelyon17, Caltech, STL-101, Stanford Cars, and Oxford pets dataset show improved classification performance compared to the CE-conv3 approach [16] and to Resnet baselines with same number of parameters.
weaknesses: * The claim of improved equivariance by order of magnitude over [16] on the 3D shape dataset is true but the error of [16] is already very low on this dataset, 0.05 at most for CEconv-3. As the error is very small, this difference in magnitude will not translate into drastic improvements in practice.
* [Comparison to stronger baselines] The comparison to [16] on non-synthetic datasets displays larger errors than in [16]. Why not use their experimental protocol that seems well detailed in their paper? 
[Incomplete comparison] Only results with CE-conv3 and CE-conv4 appear in the current paper, while [16] often shows better results with CE-conv2.
* [No comparison to SOTA] Further discussion on the Camelyon-17 results would be interesting: Looking at the leaderboard on this dataset, https://wilds.stanford.edu/leaderboard/ , we note a number of approaches are performing better than the proposed approach. What would be the pros and cons of using the proposed approach vs the SOTA ?       
* The cardinality N of the group used in the experiments is not discussed. In addition, it is only introduced on page 7, it would help the reader to define the notation before as all figures are using it.

---

confidence: 3
contribution: 2
initial_rating: 3
presentation: 3
questions: 1. Fig. 1: Baseline comparison with CEConvs

2. Fig. 4
    - What is the “Model sample efficiency”? 
    - How can the error improvement be computed?
    - Fig. 4 shows that there is no improvement over Z2CNN if 10% or 100% of training example is used. Why does it happen?
    - On the other hand, the advantage of the proposed methods are shown effective when only a little portion of the training example is used. What does this mean?

3. Table 1: Baseline comparison with CEConvs

4. Fig 5: Baseline comparison with CEConvs

5. Fig 7: Baseline comparison with CEConvs

6. What is the difference between the global hue-shift out-of-distribution (A/B) and the local hue-shift out-of-distribution (A/C)?
soundness: 3
strengths: - Interpretable visual illustrations of the results
- Interesting motivation
- Decent paper writing
summary: The authors propose GCNNs which are equivariant to color variation. GCNNs have been widely used for geometric transformations while only a few research has attempted to apply GCNNs to color-equivariant networks to achieve generalization to out-of-distribution samples. Throughout the paper, the authors reasonably justified the reason for designing the color-equivariant networks and proposed a proper method for dealing with such situations. Both quantitative and qualitative evaluations are conducted in the paper.
weaknesses: Overall, it is clear to me that the proposed method is better than the baselines (ResNet50, Z2CNN) under the generalization testing setup. However, the improvements over CEConvs look marginal (performance-wise and method-wise). Other weaknesses I have found are:

1. Related Works: 
    - What is the connection/difference of the proposed setting with Domain Generalization?
2. Missing baselines to support the argument in L44-50.
    - converting input image to grayscale 
    - data augmentation

Meta-review: This submission introduces a colour-equivariant group convolutional network (GCNN) that generalises equivariance beyond the hue channel to saturation in RGB images. The proposed method employs a lifting layer capable of directly transforming input images and further leverages the geometric structure in hue-saturation-luminance (HSL) colour space. Experiments on synthetic and real-world datasets show improved classification performance over baseline approaches.

This paper was discussed at length with the SAC. After reviewing the paper, rebuttal and resulting discussions AC believes that the overall strengths outweigh the weaknesses and recommends acceptance. For the camera-ready version, the authors should incorporate all key rebuttal results and discussion points as suggested.

Decision: Accept (Poster)