A Hypertoroidal Covering for Perfect Color Equivariance

Yulong Yang; Zhikun Xu; Yaojun Li; Christine Allen-Blanchette

cs.CVMar 4, 2026

A Hypertoroidal Covering for Perfect Color Equivariance

Authors: Yulong Yang, Zhikun Xu, Yaojun Li, Christine Allen-Blanchette

Organizations: Princeton University · Tsinghua University

Abstract

When the color distribution of input images changes at inference, the performance of conventional neural network architectures drops considerably. A few researchers have begun to incorporate prior knowledge of color geometry in neural network design. These color equivariant architectures have modeled hue variation with 2D rotations, and saturation and luminance transformations as 1D translations. While this approach improves neural network robustness to color variations in a number of contexts, we find that approximating saturation and luminance (interval valued quantities) as 1D translations introduces appreciable artifacts. In this paper, we introduce a color equivariant architecture that is truly equivariant. Instead of approximating the interval with the real line, we lift values on the interval to values on the circle (a double-cover) and build equivariant representations there. Our approach resolves the approximation artifacts of previous methods, improves interpretability and generalizability, and achieves better predictive performance than conventional and equivariant baselines on tasks such as fine-grained classification and medical imaging tasks. Going beyond the context of color, we show that our proposed lifting can also extend to geometric transformations such as scale.

Figures & tables

Figure 1 : Hue, saturation, and luminance lifting. We lift an input image with respect to the hue, saturation, and luminance channels. Hue lifting of $\mathbb{T}^{3}$ CEN, which follows the hue lifting proposed in Yang et al. (2024) . Saturation and luminance lifting of $\mathbb{T}^{3}$ CEN using a double-cover to give cyclic behavior to the saturation and luminance group. A hue, saturation, or luminance shifted input yields a cyclically permuted lifted representation.

Figure 2 : $\mathbb{T}^{3}$ CEN and LCER feature maps under HSL shifts. The features maps of $\mathbb{T}^{3}$ CEN are equivariant to shifts in hue, saturation, and luminance, while the feature maps of LCER are only equivariant to shifts in hue. (a) The images are related by a $90^{\circ}$ hue rotation. (b) The images are related by a $0.5$ shift in saturation. (c) The images are related by a $0.5$ shift in luminance. In all cases, because our $\mathbb{T}^{3}$ CEN network is equivariant to color shifts (i.e., hue, saturation, and luminance shifts), the feature maps transform predictably (cyclically permuted). Conversely, LCER is only equivariant to hue shifts.

Figure 3 : Saturation equivariance error. The normalized saturation equivariance error for

\mathbb{T}^{3}

CEN and LCER is reported.

\mathbb{T}^{3}

CEN has average error

4.66\times 10^{-6}

compared to LCER at

0.445

Figure 4 : Lifting error comparison. An input images is lifted to the respective saturation group, shifted down by $0.75$ , and shifted up by $0.75$ . We compare the restored and original input image for $\mathbb{T}^{3}$ CEN ( top ) and LCER ( bottom ). The average 8-bit integer RGB error is $6.33\times 10^{-6}$ for $\mathbb{T}^{3}$ CEN and $8.65$ for LCER.

Network	A/B	A/C	Param
ResNet44	51.25	26.66	2.6M
CEConv-3	0.02	0.05	3.8M
CEConv-4	0.60	0.53	4.9M
LCER-H3	0.03	0.04	2.6M
LCER-H4	0.00	0.00	2.6M
$\mathbb{T}^{3}$ CEN-H3	0.03	0.04	2.6M

Table 1 : Generalization to hue shift. Classification error on the 3D Shapes datasets is reported. The models are evaluated on in-distribution (A/A), global out-of-distribution (A/B) and local out-of-distribution (A/C) test sets.

\mathbb{T}^{3}

CEN performs on par with LCER and achieves improved generalization performance over ResNet and CEConv.

Network	A/B	A/C	Param
ResNet44	41.40	42.20	2.6M
LCER-S3	0.00	0.04	2.6M
$\mathbb{T}^{3}$ CEN-S3	0.00	0.00	2.6M

Table 2 : Generalization to saturation shift. Classification error on the saturation shifted 3D Shapes datasets is reported. The models are evaluated on in-distribution (A/A), random saturation shifts (A/B), and component-wise random saturation shifts (A/C) test sets.

\mathbb{T}^{3}

CEN achieve improved generalization performance over LCER and ResNet.

Network	A/A	A/B	A/C	Parameter
ResNet-18	8.32	37.70	33.88	11.2M
LCER-L3	7.31	34.83	36.43	11.1M
$\mathbb{T}^{3}$ CEN-L3	5.36	14.42	31.53	11.2M
$\mathbb{T}^{3}$ CEN-L4	6.60	13.09	29.53	11.2M
$\mathbb{T}^{3}$ CEN-L8	5.67	11.88	27.46	11.1M
$\mathbb{T}^{3}$ CEN-L16	5.66	11.09	24.32	11.2M

Table 3 : Generalization to luminance shift. Classification error on the small NORB dataset is reported. The models are evaluated on medium lighting (A/A), low lighting (A/B), and bright lighting (A/C) test sets.

\mathbb{T}^{3}

CEN achieves improved generalization performance over LCER and ResNet. Moreover, with the same order,

\mathbb{T}^{3}

CEN-L3 outperforms LCER-L3.

Network	Error	Param
ResNet44	55.40 (2.19)	2.6M
LCER-H4S3	9.76 (3.54)	2.6M
$\mathbb{T}^{3}$ CEN-H3S3L3	3.01 (1.61)	2.6M
$\mathbb{T}^{3}$ CEN-H4S4L4	0.00 (0.00)	2.6M

Table 4 : Generalization to HSL shift. Classification error on the HSL shifted 3D Shapes dataset is reported.

\mathbb{T}^{3}

CEN achieves improved generalization performance of LCER and ResNet.

Network	Error	Param
ResNet50	28.91 (7.58)	23.5M
CEConv-3	28.76 (9.93)	23.1M
LCER-H4	27.53 (3.39)	23.5M
LCER-S3	16.08 (2.68)	23.3M
LCER-H4S3	19.06 (4.92)	23.0M
$\mathbb{T}^{3}$ CEN-H4	27.53 (3.39)	23.5M

Table 5 : Generalization to color shift in Camelyon17. Classification error on the Camelyon17 dataset is reported.

\mathbb{T}^{3}

CEN achieves improved generalization performance over LCER and ResNet.

Saturation Shifted Dataset
Network	Caltech 101	CIFAR-10	CIFAR-100	Stanford Cars	Oxford Pets	STL-10
ResNet	56.29 (0.59)	11.71 (1.16)	47.58 (5.19)	37.34 (3.75)	57.92 (2.49)	30.28 (0.76)
ResNet-Jitter	49.04 (0.63)	11.91 (0.58)	43.13 (2.13)	32.00 (6.41)	51.88 (1.05)	28.86 (0.40)
ResNet-AugMix	35.53 (2.50)	11.80 (0.61)	46.93 (0.95)	30.21 (0.92)	44.84 (1.11)	23.57 (2.36)
ResNet-DeepAug	42.87 (0.47)	20.39 (1.44)	51.64 (0.60)	42.32 (1.11)	43.57 (1.65)	24.17 (1.49)
ResNet-Plackian	38.05 (2.99)	11.87 (1.24)	47.12 (1.89)	33.48 (3.16)	41.43 (4.20)	23.69 (2.79)

Table 6 : Generalization to color shift. Classification error on the Caltech-101, CIFAR-10, CIFAR-100, Oxford-IIT Pets, Stanford Cars, STL-10 datasets are reported.

\mathbb{T}^{3}

CEN achieves improved generalization performance over baselines on saturation and luminance shifted testsets, where saturation and luminance were reduced by

0.5

. Performance on the vanilla datasets are reported in Table 7 where

\mathbb{T}^{3}

CEN performs on par with LCER and outperforms baselines. See Appendix D for details on dataset and training.

Figure 5 : Lifting coverage and cases of redundant representations. (Top) Coverage of the representation

f^{0}=\varphi_{hsl}(\cdot,x)

for different input saturation and luminance values. The original input value and the first, second, and third lifted representation is in black, blue, orange, and green. (Bottom) Information entropy of the

f^{0}

. Redundant representations are highlighted with red lines.

Figure 6 : Lifting entropy density. We show the entropy density (information entropy divided by order) at different order and input saturation and luminance values. The order with the highest entropy density for every input is highlighted with red boxes.

Figure 7 : $\mathbb{T}^{3}$ CEN equivariant feature map under RGB shift. To show the group actions are equivariant we qualitatively test for commutativity for both LCER and $\mathbb{T}^{3}$ CEN. We demonstrate that the feature maps of (a) $\mathbb{T}^{3}$ CEN is equivariant to shifts to the RGB channels while that of (b) LCER is not.

Figure 8 : $\mathbb{T}^{3}$ CEN equivariant feature map under scale. To show the group actions are equivariant we qualitatively test for commutativity for both LCER and $\mathbb{T}^{3}$ CEN. We demonstrate that the feature maps of (a) $\mathbb{T}^{3}$ CEN is equivariant to shifts in scaling while that of (b) LCER is not.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 9 : Degradation of performance with large lifting cardinality. (Left) We report the classification accuracy with increasing hue cardinality on the hue shift MNIST dataset ( LeCun et al., 2002 ) (for more details see Appendix D.5 ). With increasing cardinality, the classification accuracy drops. (Right) We report the number of channels in the final feature map with increasing hue cardinality on the Z2CNN architecture ( Cohen and Welling, 2016 ) . With increasing cardinality the number of channels drops significantly.

Figure 10 : Saturation equivariance error. The normalized saturation equivariance error for

\mathbb{T}^{3}

CEN and LCER is reported.

\mathbb{T}^{3}

CEN has average error of

4.66\times 10^{-6}

while LCER has average error of

0.445

Original Dataset
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)
ResNet-Gray	33.79 (3.09)	8.45 (0.68)	32.04 (0.66)	24.71 (0.93)	30.38 (0.35)	18.71 (1.47)
ResNet-Jitter	32.90 (0.82)	8.33 (0.44)	32.27 (0.18)	22.38 (1.65)	30.06 (0.52)	17.89 (1.48)
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)

Appendix

Table 7 : Generalization to color shift. Classification error on the Caltech-101, CIFAR-10, CIFAR-100, Oxford-IIT Pets, Stanford Cars, STL-10 datasets are reported.

\mathbb{T}^{3}

CEN performs similarly to CEConv and LCER on the vanilla test datasets.

	Caltech 101	CIFAR-10	CIFAR-100	Stanford Cars	Oxford Pets	STL-10
Full	0.80%	0.57%	0.69%	0.88%	0.65%	0.89%
Partial	0.69%	0.64%	0.64%	0.67%	0.48%	0.66%

Appendix

Table 8 : Redundant lifting metric. We report the percentage of pixels in the datasets in Table 6 that exhibit full or partial redundant values.

	Caltech 101	CIFAR-10	CIFAR-100	Stanford Cars	Oxford Pets	STL-10
Arch.	ResNet18	ResNet44	ResNet44	ResNet18	ResNet18	ResNet18
Batch Size	16	128	128	16	16	16
Epoch	300	300	300	300	300	300
Optimizer	Adam	SGD	SGD	Adam	Adam	Adam
Learning Rate	$10^{-2}$	$10^{-1}$	$10^{-1}$	$10^{-2}$	$10^{-2}$	$10^{-2}$
Scheduler	N/A	cos anneal.	cos anneal.	N/A	N/A	N/A

Appendix

Table 9 : Training hyperparameters. We report the training hyperparameters on the Caltech-101, CIFAR-10, CIFAR-100, Oxford-IIT Pets, Stanford Cars, STL-10 datasets. All datasets are trained under cross entropy loss. All datasets are training by replacing conventional convolution blocks with

\mathbb{T}^{3}

CEN convolution. The filter count for

\mathbb{T}^{3}

CEN is reduced to match that of the respective vanilla ResNet. Group pooling is performed at the last layer to obtain an invariance representation for classification. LCER and CEConv are training using network parameters provided in Lengyel et al. (2023) and Yang et al. (2024) respectively.

Panel of Figure 11 — Figure 11 : Hue and saturation shifted 3D shapes. Examples images from the (a) hue and (b) saturation shifted 3D shapes dataset.

Figure 12 : Small NORB dataset. (Left) The train set and in-distribution test set $A$ with medium lighting conditions; (Middle) The first out-of-distribution test set $B$ with low lighting conditions; (Right) The second out-of-distribution test set $C$ with high lighting conditions.

Figure 13 : Hue shift MNIST dataset. (Left) The train set hue in range of $0^{\circ}-120^{\circ}$ and (Right) the test set hue in range of $120^{\circ}-360^{\circ}$ .

Panel of Figure 14 — Figure 14 : Camelyon17 dataset. We show (a) example slides from the Camelyon17 dataset as well as (b) saturation distribution.

	H1	H4	S4	L4	H4S4	H4L4	H4S4L4
Training Memory (MiB)
$\mathbb{T}^{3}$ CEN	1030	2739	2739	2739	9480	9480	32842
LCER	1014	2702	2702	2702	9906	9906	not supported
Training Speed (it/s)
$\mathbb{T}^{3}$ CEN	19.38	11.33	11.34	11.27	5.23	5.24	1.59
LCER	19.01	11.30	11.31	11.31	5.22	5.21	not supported

Appendix

Table 10 : Training computation expense. We report the training iterations per second and memory consumption, where

\mathbb{T}^{3}

CEN. Training memory consumption is obtained from nvidia-smi . Training time is reported using tqdm . For both metrics we discard the first 100 iterations to account for model loading. We report the value averaged over the next 100 iterations for the ResNet18 backbone. LCER does not support lifting to both the saturation and luminance channel.

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Normalization Equivariance for Arbitrary Backbones, with Application to Image Denoising

Youssef Saied, François Fleuret

Department of Computer Science, University of Geneva, Switzerland · Meta FAIR.

Rotation Equivariance Batch Normalization Iterative Denoising Methods Normalization

Open paper AskPDF arXiv

Normalization Equivariance (NE) is a structural prior that improves robustness to distribution shift in image-to-image tasks. A function

f

is normalization equivariant iff

f(a y + b\mathbf{1}) = a f(y) + b\mathbf{1}

for all

a>0

and

b\in\mathbb{R}

. Existing NE methods constrain every internal layer to NE-compatible operations. These constraints add runtime cost and exclude standard transformer components such as softmax attention and LayerNorm. We introduce Wrapped Normalization Equivariance (WNE), a parameter-free wrapper that normalizes the input, applies any backbone, and denormalizes the output. We prove every NE function admits this factorization, so the wrapper exactly parameterizes the class of NE functions. On blind denoising, wrapping CNN and transformer architectures improves robustness under noise-level mismatch with no measurable GPU overhead, while architectural NE baselines are up to

1.6\times

slower.

Abstract

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