Designing Affine-Invariant Neural Networks for Photometric Corruption Robustness and Generalization
Authors: Mounir Messaoudi
Quentin Rapilly
Sébastien Herbreteau
Anaïs Badoual
Charles Kervrann
Venue: ICLR 2026 Poster (2026)
Keywords: Computer Vision
Robust neural network
invariance
equivariance
biological imaging
microscopy
classification
object localization
Primary area: applications to computer vision, audio, language, and other modalities
TL;DR: Scale-Equivariant & Shift-Invariant NN rendered Affine invariant for application to image classification and object localization

Abstract: Standard Convolutional Neural Networks are notoriously sensitive to photometric variations, a critical flaw that data augmentation only partially mitigates without offering formal guarantees. We introduce the *Scale-Equivariant Shift-Invariant* (*SEqSI*) model, a novel architecture that achieves intensity scale equivariance and intensity shift invariance by design, enabling full invariance to global intensity affine transformations with appropriate post-processing. By strategically prepending a single shift-invariant layer to a scale-equivariant backbone, *SEqSI* provides these formal guarantees while remaining fully compatible with common components like ReLU. We benchmark *SEqSI* against *Standard*, *Scale-Equivariant* (*SEq*), and *Affine-Equivariant* (*AffEq*) models on 2D and 3D image-classification and object-localization tasks. Our experiments demonstrate that *SEqSI* architectural properties provide certified robustness to affine intensity transformations and enhances generalization across non-affine corruptions and domain shifts in challenging real-world applications like biological image analysis. This work establishes *SEqSI* as a practical and principled approach for building photometrically robust models without major trade-offs.

Published as a conference paper at ICLR 2026
DESIGNING AFFINE-INVARIANT NEURAL NETWORKS FOR PHOTOMETRIC CORRUPTION ROBUSTNESS AND GENERALIZATION
Mounir MESSAOUDI", Quentin RAPILLY"', Sébastien HERBRETEAU",
Anaïs BADOUAL? and Charles KERVRANN
1 Inria Center at University of Rennes, France, SATRPICO Team
U1339 INSERM, Institut Curie, Chemical Biology of Cancer Unit, Paris, France
(mounir messaoudi, quentin rapilly, anais badoual, charles kervrann aeinriannnnnn...... fr
2 Univ Rennes, Ensai, CNRS, CREST-UME 9194, F-35000 Rennes, France
corresponds equal contribution.
ABSTRACT
Standard Convolutional Neural Networks are notoriously sensitive to photometric variations, critical flaw that data augmentation only partially mitigates without offering formal guarantees. We introduce the Seale-Equivarians Shift- Invariann (SEqSI) model, novel architecture that achieves intensity scale equivariance and intensity shift invariance by design, enabling full invariance to global intensity affine transformations with appropriate post-processing By strategically prepend- ing a single shift-invariant layer to scale-equivariant backbone, SEqSI provides these formal guarantees while remaining fully compatible with common com- ponents like ReLU. We benchmark SEqSI against Standard, Scale houle-Equvvaiiat (SEq), and and object-localization tasks. Our experiments demonstrate that SEqSI architec- tural properties provide certified robustness affine intensity transformations and enhances generalization across non-affine corruptions and domain shifts in chal- lenging real-world applications like biological image analysis. This work estab- lishes SE&SI a practical and principled approach for building photometrically robust models without major trade-offs.
(AffEq) models on 2D and 3D amage-classificattee
INTRODUCTION
Despite their widespread success, standard Convolutional Neural Networks (CNNs) are notoriously sensitive to photometric variations. Semantically irrelevant changes in image contrast or brightness can drastically alter model predictions (Guan & Liu 2022 Hendrycks & Dietterich 2019 Torralbal & Efros 2011). a critical flaw for real-world applications. Data augmentation, mitigates this issue but acts brute-force solution, offering formal guarantee of robustness.
In this work, we argue for more principled approach: encoding robustness directly into the network architecture. We introduce the Scale-Equivariaa Shift-Invariant (SEqSI) model, novel architec- ture whose outputs are provably invariant to global intensity shifts and equivariant to global intensity scales by construction. This design provides robust foundation that can be rendered fully invariant to affine intensity transformations with appropriate post-processing. Crucially, SE- qSI achieves this guarantee while remaining practical; is compatible with standard activations and introduces negligible computational overhead. Our main contributions are the following ones:
We propose novel architecture (SEqSI) which is by design intensity Scale Equivariant and intensity Shifi invariant.
We address the challenge of achieving invariance from networks whose logits are either affine-equivariant (i.e., scale+shift equivariant) or scale-equivariant and shift-invariant. We
We call logits the raw output of the network, before any task-specific post-processing.

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Published as a conference paper at ICLR 2026
show that while this is straightforward for argmax- based tasks (e.g.. classification), stan- dard pipelines for threshold based tasks (e.g., object localization) are incompatible with such architectures. We resolve this by introducing coherent framework pairing output standardization at inference with a novel Z-scored Mean Squared Error (ZMSE) loss.
We benchmark SEqSI against Standard. Scale Seale-Equivarians (SEq) (Mohan et al. 2019). and the more restrictive Eaiire--iivvaian (AffEq) (Herbreteau et al. 2023) models. We show that SEgSI architectural design provides certified robustness to afline transformations and enhances generalization to non-affine corruptions.
We demonstrate that, unlike normalization pre-processing which only handles global trans- formations, the architectural properties of SEqSI makes it inherently robust to range of apatially-varying affine intensity transformations.
We demonstrate the advantages of SEqSI challenging biological imaging tasks, includ- ing macromolecule classification in Eryo-Electron Tomography (Cryo-ET) and object lo- calization in fluorescence microscopy. In these two application fields, where severe and photometric shifts cause Standard models to fail, SEgSI architectural guarantees provide robust out-of-distribution generalization, while maintaining high accu-
racy where baselines collapse (see Fig. D
Whp Distribbtiion
Cififeconooved (Dut Distribbtttbb
Denolsed (Dut-oh Distribution)
Isonetcorrected (Out-at-Distributtn
This paper is organized as follows. In Section 2 we give formal defini- tions. Section 3] we present re- lated work. In Section 4 we de- scribe the SEqSI architecture and its theoretical guarantees. Furthermore, we present an extensive experimental validation, starting with classification benchmarks (Section 5). demonstrat- ing higher nut-of-distribution gener- alization capabilities on challenging cryo-ET data (Section (6). and con- cluding with its application to object localization (Section)
SEgEi Standard
87.17%
48.91% # 11.04%
22.36%
15.95% = 1..4%
1.50%
6.59%
85.15%%
66.51% 99.75%
74.53% 4.25%
3.59%
= 4.18%
Figure 1: SEqSI is robust to real-world domain shifts for macromolecule classification Cryo-ET (CZJ) data. Trained on one domain, Weighted Back- Dack-Projection (WBP), and tested on three unseen ones, SEqSI generalizes to all unseen domain, while the Standard model collapses. Mean accuracy (% = std over 5 seeds) is color-coded (green: success, red: fail).
PRELIMINARIES
In this section, we present our framework: the group of photometric transformations of interest and the key properties of equivariance and invariance, which are the mathematical basis for building networks with provable robustness. Let us define network an input image x E X' to an output E Y. also called logits. The input space X' encompasses 2D or 3D images, which can be single or multi-channel. In the following, we focus on the large family of CNN, and introduce constraints that ensure desired properties. In the paper, we refer to an unconstrained architecture layers (either fully connected or convolutional layers, + Wx x - b, where W is a weight matrix and b an additive bias) and non-linear activations.
function f X
that maps
Standard network, commonly designed as the composition of linear
Photometric transformations. These corruptions modify the intensities values of an image without altering its spatial structure. photometric transformation is global if the same operator is applied to every pixel. This operator may be be non-linear, such as the gamma correction (x x"). For spatially-varying (local) transformations, the intensity variation depends on pixel coordinates (e.g., non- -inn-orioom illumination). As shown by Hendryykk & Dietterich (2019), these variations can significantly degrade CNN performance.
Our work focuses the group of global affine intensity transformations, which apply uniform scaling and shifting operation to every pixel, modeling changes in contrast and brightness. This group, denoted Tanr is composed of scaling (Tacate) and shifting (Tana) operations:
Tal (Tx,p + Xx + j X E Ri IR),
(1)
where Taaale = {ax= x + Aæ X e ] and Tain = {b, : x + x + I l R). The condition > 0 ensures that the order of intensities is preserved. While theoretical guarantees apply
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specifically Tan, also empirically evaluate robustness against broader set of photometric corruptions.
Equivariance and Invariance. These notions are fundamental properties for designing robust ma- chine learning models, especially in computer vision. They describe how a function output be- haves under input transformations: equivariance implies predictable transformation of the output, whereas invariance means that the output remains unchanged. We formally define these concepts bellow.
Let ʃ X' + y/ be function (e.g., a neural network) and T be a group of transformations defined on both, X' and y. The function is:
T -equivarrant if, for any T' T, applying the transformation before or after the function f yields the same results: AT' E T, f(T(=)) = T(f())).
T-invariant if, for any T' T, the output of the function remains unchanged after applying the transformation to the input: VT E TT,((((a)) - f(x).
RELATED WORK
Approximate Robustness. Common strategies for robustness include data augmentation (Krizhevsky al., 2012, Hendrycks et al., 2020) and regularization techniques that penalize in- consistencies under transformations in their loss function (Chen et al. [2022] Midtvedt et al. (2022). While effective, these methods only encourage approximate invariance and offer no formal guaran- tee, which is the main contribution of the paper.
Robustness by Architectural Design. A more principled approach. which we adopt, is to de- sign architectures that are equivariant by construction, building on the theory of -oouu--quivariiant CNNs (Cohen & Welling 2016). For photometric transformations, various categories of work ex- ist. Some focus on equivariance to color changes (Lengyel et al. [2033] Yang et al. 2024). Others, like our work, address intensity variations with Scale- Eaull--Euivaiannt (SEq) networks (Mohan et al.) 2019) and, more recently, proaches build on a few key principles. SEq networks are constructed by simply removing biases from all layers, making them equivariant scaling and compatible with standard activations like ReLU. To achieve full affine equivariance, the more restrictive AffEq model additionally constrains convolutional weights sum to one and replaces standard activations with specialized 'SortPool' layer. 'SortPool' operates on pairs of features (C1, C2) by sorting their values at each spatial location:
Equivaraquv (AffEq) networks (Herbreteau al. 2023). These ap-
SortPool: (c1, e2) + (min(c1, c2), max(c1, 22)). While this guarantees full affine equivariance, introduces significant trade-off: constraints on weights and activations make AffEq computation- ally expensive.
4 METHOD
Our goal to design neural networks whose final predictions are provably invariant to global affine photomet- ric transformations (Tar). Unlike standard approaches, which rely on fragile input normalization step, we build robustness directly into the model architecture. Our core strategy a two-step process: first, we design network whose output logits, fxx)
Table 1: Architectural constraints for the four model families.
These principles can be applied to any standard backbone. Weights constraint are applied channel-wise.
Model
Activation
Bias
Weights constraint
Standard
ReLU
Yes
None
SEq
ReLU
No
None
AffEg
SortPool
No
Σ (for all layers)
SEqSI (ours)
ReLU
w = (Ist layer only)
are shift invariant and scale equivariant by design; second, we apply simple, task-specific post- processing function these logits to guarantee final invariant prediction. This principled approach
ensures robustness by construction, as detailed in the following sections.
Constraints applied to convolutions apply analogously linear layers.
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SortPool o AffEq Convolution
ReLU
ReLU
000 SEq
Convolution
Affine Equivariant Convolution
SEqSi Convolution
Scale Equivariant Convolution
)=
weight restrictions
Blas Free
Free
Bias Free
(a) Affine- Afftne-EEuivaateee Equivariant (AffEq) block.
(b) Scale- Baaev--quvvaiiant Shift- Invariant (SEqSI) block.
Figure 2: Comparison of equivariant architectural strategies. (a) The AffFa model from Herbreteau et al. (2023) enforces strict equivariance in every layer, requiring constrained convolutions and specialized "Sort- Pool activation. (b) Our proposed SEqSI architecture establishes robustness by prepending single shift- invariant convolution to backbone scale-equivariat layers (bias-free convolution with ReLU). This design simplifies the construction of robust networks with minimal constraints.
4.1 PROPOSED ARCHITECTURE
We introduce the Scale- Equivaraua Shift-Invariant (SEqSI) model (Fig. 2 practical and efficient architecture that achieves robustness through minimal, strategic design. Its construction follows two principles, following Proposition 2 (see Supp. B.I). First, to ensure scale-equivariance, all lay- ers in the network are made scale-equivariant by using bias-free layers, scale-equivariant activations like ReLU, and standard pooling layers, which are inherently affine-equivariant (see Supp. [B.3). Second, to achieve invariinn,,,,nce, the initial layer is also made shift-invariant by constraining its weights to sum to 2 As shift-invariance is established by this single initial layer, the rest of the network remains a standard scale-equivariant backbone, while keeping compatible with high- performance components. This composition yields logits that are provably scale-equivariant and shift-invariant (f(xæ + 44) = >f())), providing robust foundation for achieving full affine in- variance with an appropriate task specific post-processing. In Supp. E.3.2 we even show that our approach provide weak invariance guarantees for spatially varying affin transformations such as piecewise constant intensity shifts or intensity shifts varying linearly in space.
The design of SEgSI is motivated by the end-task. While tasks like denoising require full equiv- ariance every layer, which forces the use of restrictive architectures like AffEq, classification or localization require final invariance. By establishing shift-invariance at the very first layer, SEgSI frees the following layers of the network from strict shift- equivariiie constraints, making our ap- proach more practical. The constraints for all benchmarked models are summarized in Table This avoids the significant computational overhead of AffEq, which is over 50% slower and requires 3x more memory. SEqSI adds theoretical guarantees than the Standard baseline with an identical memory footprint (see Supp.[C.I). Furthermore, unlike AffEq, SEqSI is compatible with transfer learning from Standard architectures (see Supp. C.10).
computational cost and trains 5% faster
Supp.
4.2 TASK SPECIFIC POST-PROCESSING
Post-processing for Argmax-based tasks. For any task for which the final prediction is derived from an 'argmax' operation on the logits such as classification or segmentation achieving affine invariance is straightforward. The core principle is that the 'argmax' operation is inherently in- variant to any transformation that preserves the order of its inputs. Both models, SEqSI and AffEq are designed such that an affine transformation on the input image induces an order- pnder-preserving transformation on the output logits. Consequently, the 'argmax' of the transformed logits remains unchanged, yielding an invariant prediction. This property proven in Supp. C.3 holds even when a strictly monotonic increasing function (such as 'softmax") is applied before the "argmax' operation.
Post- post-processing for tasks requiring thresholding. The classical way to solve object localiza- tion consists in converting the logits score-map é, generally via a sigmoid function (to scale results in [0, 1)). The network is trained to generate local maxima of this map where objects are located. During inference, the maxima exceeding a chosen threshold are considered as object lo- cations. This thresholding step is incompatible with an equivariant network; affine transformation
"To our knowledge, scale-invariance of a convolution is impossible to obtain (see Supp- B.2 WWe call "weak invariance" result that fully invariant, e.g., invariant in image sub-parts only.
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on the input image produces transformed logits whose range is shifted and/or scaled, rendering any fixed threshold meaningless and breaking end-to-end invariance.
We address this issue by introducing an alternative way to generate the score-map that becomes in- variant to affine transformation when paired with an equivariant network. consists in standardizing the network output, follows:
y
13 1
2 (
(2)
(1)
where [[]] and (.) denote the mean and standard deviation (std) operators, respectively. The thresh- olding of & 2 y consists in considering only the voxels whose score is higher than the mean by times the std of the entire map. In Supp. we provide the invariance proof and further consider- ations of this thresholding strategy.
The usual training strategy dedicated to object localization, based on the Binary Cross Entropy (BCE) loss function, requires input standing in the range [0,1]. Hence, it is incompatible with our new score map generation. Accordingly, we introduce a new loss function, named Z-scored Mean Squared Error (ZMSE):
=( MSE 2, 2()) MSE 2()).
where is the ground-truth (GT) score map. The GT is usually defined by assigning to the background (outside the objects), and values decreasing linearly from at the object's barycenter to O at its boundaries. The ZMSE forces the network to learn the relative spatial distribution of score rather than the absolute values, and do not impose the distributions to lie in the same range, as we are only interested in saving the order and locating maxima. This strategy can be adapted other tasks requiring a thresholding such as binary segmentation.
All experiments presented in this paper are repeated with different seeds to ensure statistical significance, resulting in 5 independently trained model instances.
5
EXPERIMENT 1: IMAGE CLASSIFICATION
In this section, we demonstrate the robustness of SEqSI to photometric corruption for image classi- fication.
5.1 EXPERIMENTAL. SETUP
| based on a ResNet-20 backbone (He al. 2015), presented in Supp. C.T Standard, SEq, AffEq. and our proposed SEqSI models. For the SEqSI and AffEq models, an invariant prediction obtained by applying a 'softmax' followed by an 'argmax' operation to the logits, as detailed in Section 4.2
Architectures. We benchmark four model families summarized in Table
Dataset and Training. All models are trained on CIFAR-10 (Krizhevsky et al., 2009). We select the best checkpoint based on validation accuracy and report the mean performance across the runs (see Supp. Table I0 for full results). To isolate architectural contributions to robustness, our default training pipeline uses only geo- metric data augmentation (e.g., random flips and crops), deliberately excluding any photometric augmentations. This ensures that robustness to photometric corruptions a direct result architectural design, not training data (see details in Supp. C.I
Table 2: Summary of empirical invari- ance tests Supp. Table 7 model is marked as invariant (v) if it achieves 0% prediction error for all transformations within a class (Shift, Scale, Affine).
Model Standard SEq SEgSI AffEq
Shift
Scale
Affine
X
x
v
X
V
5.2 RESULTS AND DISCUSSION
Certified Invariance. We begin by empirically validating the theoretical properties of the models by measuring their prediction invariance error the percentage of predictions the CIFAR-10
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Table Robustness corruptions on the CIEAR-10 set. Test accuracy (%) of four architectures under various photometric comuptions. Models are grouped by the training augmentation strategy used: none (0), aftine (Aff.), non-affine (NAff.), ,, or all combined (All). Each row corresponds to different perturbation applied evaluation. Within each training strategy (column group), the hest result is highlighted in gray. The overall hest accuracy for each perturbation in bold. Full results with standard deviations are given Supp. Table
Train Aug.
0
Train Ang. = Aff
Train Aug. = NAIf
Train Aug
All
Corruption
Model
Sund.
SEq
SBçS1
Allly
Stand.
SEg
SEqS1
Alllg
Stand.
SEq
SEqSI
AlD
Stand.
SEq
SBgSI
Afliq
Original
91.7
91.3
91.2
89.6
91.2
9016
91.0
38.9
92.1
92.1
91.5
90.5
92.0
91.6
91.8
90.6
Affine transformations (Aff.)
Shift
51.1
53.4
91.2
89.6
91
90.3
91.0
88.9
58.5
65.7
91.5
90.5
91.4
90.7
91.8
90.6
Scale 1)
68.0
9913
91.2
89.6
90.3
90.6
91.0
88.9
55.8
92.1
91.5
90.5
90.6
91.6
91.8
90.6
Scale 1)
91.0
92.1
91.5
90.3
92.0
64.3
Non-Afline transformations
Scale 1) saturated
78.1
77.6
87.1
saturated
59.0
72:1
72.6
71.7
70.6
84-8
84.1
82.8
Spurially-variinn Affine
87.0
89.5
88.6
Contras: Inversion
15.4
56.5
$2.9
61.3
56.2
$2.9
55.0
90.2
90.3
87.5
85.3
39.2
57.8
85.5
84.6
Nosse (low)
90.
90.6
90.1
88.5
9000
$91.
50.00
87.8
91.6
91.7
91.0
900
91.5
91.2
91.3
89.8
Noise (high)
19.0
22:4
15.2
20.3
15.5
19.8
16.0
19.2
79.6
80.0
78.7
76.6
78.3
78.
77.9
75.2
Gamma (darken)
74.3
76.4
79.9
76.4
K2.5
79.
79.9
76.2
89.6
13.8
88.3
85.6
89.5
57.8
88.0
96.0
Gamma (Rigtecs
85.9
85,3
90.3
88.6
90.6
89.9
90.0
38.3
91.8
91.9
91.1
90.2
91.7
91.3
91.5
90.2
(b)
(d)
(e)
(g)
Original
Snift
Shift Saturated
Scale (<1)
Scale (=1)
Scale (>1] Saturateo
Affine
(h)
(k)
(
(m)
Spatially-varying Mfli Contrast Inversion
(low)
(high)
(lighten)
(darken)
Affine Saturated
Figure 3: Photometric corruptions used for model robustness evaluation and data augmentation. The figure displays reference image (a) and its alteration by various types of corruptions. The distribution plot above each represents the reference pixel distribution (blue) and the perturbed pixel distribution (red). Each sub- figure is labeled for reference: (all transformations, except (h), are global: a unique value is randomly picked in the corresponding distribution) additive shift with E [-2.0, 2.0] (b) and saturated version with μ [-0.7, 0.7] (c): scaling with À E [0.0, 1.0 (d), À E [1.0, 1.0] (e), and saturated version with À E [1.0,3. 3.0] (f); affine transformation with E [0.0, 4.0), E [-2.0, 2.0] (g): a spatially varyinn affine transformation with A(u, v) E [0.1, 0.5) and uu,,, v) [-1, where u.v are pixel coordinates (h); contrast inversion which is spatially-varying scale transformation with X(u, u) E [-1.0, -0.2] (i): additive Gaussian noise I T(x)) ) x
where (N, a²) with low intensity (a E 10.0, 0.03)) (i) and high intensity (a [0.15, 0.255) (k); non- linear transformations via gamma correction T(x) x", showing image lightening (r E [0.2, 1.0) ()) and
darkening (y E (1.0,5.0) 5.0)) (m): and affine saturated with À 0.7, 1.3), 3), μ (-0.3, 0.3] (n);
test set that change when global affine intensity transformation is applied (protocol in Supp. C.4 As summarized in Table and detailed in Supp. Table 7 both SEqSI and AffEq achieve perfect (0% error) affine invariance, empirically confirming their design. In contrast, the Standard model not invariant, and the SEq model only invariant to scaling, highlighting the effectiveness of the architectural constraints.
Architectural Robustness Affine and Non-Affine Photometric Corruptions. To investigate the impact of architectural design, we train all models without photometric augmentation (Table 3 col. Ø) and evaluate their performance against the wide spectrum of corruptions shown Fig. SE and AffEq demonstrate their advantage: they maintain high accuracy on all affine corruptions, unlike Standard and SEq models whose performance degrades significantly. Notably. this architec- tural benefit extends to unseen non-affine corruptions, where SEqSI outperforms other models on out of 9 transformations. This suggests that built-in affine invariance fosters a more general- ized form of robustness, even for unseen corruptions beyond the theoretical guarantees of the model.

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Interaction with Affine and Non-Affine Data-Angmeenatioon We next investigate the interplay between architectural priors and data-driven robustness, by training all models with three distinct augmentation strategies: Aff. (non-saturated affine), NAff. (non-affine and saturated affine), and All (the complete set of 13 perturbations). To ensure a fair comparison, the transformations used for augmentation share the same parameters those used for evaluation (Fig. 3 A detailed description of these augmentation strategies is provided in Supp. C.2 The results are presented in Table
The key finding is that architectural priors and data augmentation are complementary. In this con- trolled setting, where training augmentations perfectly match the evaluation corruptions, the uncon- strained Standard model expected to perform better. Nevertheless, the results show that SEqSI, despite its architectural constraints, still remains competitive. When trained with the All strategy, its accuracy on non-affine corruptions is very close to the Standard. This is significant result because it demonstrates that the SEqSI prior is not overly restrictive and does not compromise the capability of the model to learn by using data augmentation. Unlike the strongly constrained AffEaq model, which systematically underperforms SEqSI model, showing a clear advantage of our architecture.
Scalability to Complex Datasets. Beyond CIFAR-10, we conducted further experiments on the more challenging Oxford-IIIT Pets and Stanford Cars datasets. The results, detailed in Supp. C.8 confirm that SEqSI maintains its certified invariance and achieves competitive performance, demon- strating its applicability to complex, high-resolution tasks. Moreover, SEqSI is compatible with transfer learning from Standard architectures, property not provided by AffEq (see Supp.C.10.
Conclusion SE provides practical method for certified affine invariance, overcoming the lim- itations of prior work. Unlike AffEq, achieves this without the high computational cost or perfor- mance degradation from overly restrictive constraints. Our experiments show that SEqSI architec- tural prior provides robustness to affine intensity transformations and generalizes to unseen non-affine corruptions, even without specific data augmentation. Crucially, it complements data augmentation, maintaining certified invariance while demonstrating a learning capability compara- ble unconstrained models across a wide range of corruptions.
EXPERIMENT 2: MACROMOLECULE CLASSIFICATION IN CRYO-ET
To demonstrate the practical benefits of our approach, we evaluate its Out-O--Ditriiutiion (OOD) generalization on a challenging real-world task: macromolecule classification in cryo-electron mography (cryo-ET), which are 3D grayscale images. We use data from the recent Chan Zucker- berg Initiative (CZI) challenge (Harrington al., 2024). Each biological sample (tomogram) is processed through four different denoising pipelines (Weighted Back- Dack-Projection (WBP), Denoised, IsoNet Corrected and CTF Deconvolved), which act as distinct photometric domains. As shown in Fig. [ Supp. Fig. [6] and Supp. Table T8] these pipelines induce severe shifts in contrast and brightness, creating a challenging generalization problem and a natural testbed for OOD robustness.
6.1 EXPERIMENTAL SETUP
We perform a 6-class macromolecule classifi- cation task using 3D ResNet backbone inspired by ResNet-18 for Standard, SEq and AffEq baselines and our SEqSI model. Our experimental design directly test OOD capabil- ities: we train models on a single domain (WBP) and evaluate their performance on the three unseen domains (Denoised, IsoNet Corrected and CTF Deconvolved). For each run, the model with the highest validation accuracy is selected for testing. We use a strict tomogram-lever split (5 train, 1 val, test) to prevent data leakage. The complete experimental protocol is detailed in Supp. D.I
Table 4: Performance comparison on test set. Models were trained on WBP data. Results are reported as mean accuracy (Acc.) std. Best results for each metric are in bold.
Standard Acc. (%)
SEq Acc. (%)
SEqSI Acc. (%)
AffEq Acc. (%
Data Type
WBP (in-distrib.)
87.17 +:2 1.50 87.69
35.15
79.26 1 4.04
CTF Deconvolved
48,91 + 11.04 65.21+6.008
66.51 + 9.75 79.07 + 5.04
Denoised
22.36 + 6.59 28.42-7.66
4.25
61.79 + 5.97
IsoNet Corrected
15.95 + 1.44 16.67+00.0
73.21 46.37 + 5.98
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6.2 RESULTS AND DISCUSSION
The results in Table 4 and Fig. I highlight the critical role of built-in invariance for OOD general- ization. On the in-distribution WBP data, the SEq model achieves the highest accuracy (87.69% closely followed by the Standard model (87.17%) and our SEgSI model (85.15%) The AffEy model is also competitive. The true advantage of architectural priors becomes evident when generalizing to the unseen domains. On the Denoised and IsoNet Corrected data, the performance of both the Stan- dard and SEq models collapses to near-random chance F In contrast, SEqSI maintains high accuracy (74.53% and 73.21% respectively). demonstrating remarkable robustness. The AffEq model also proves robust, significantly outperforming the non-invariant baselines, though it does not reach the performance of SEqSI on these two domains. Interestingly, on the CTF Deconvolved domain, AffEq is the top performer (79.07%), with SEqSI (66.51%) %) and SEq (65.21 also showing strong, com- petitive performance, far surpassing the Standard model (48.91 %). This provides strong evidence that architectural invariance is a key mechanism for generalizing across real-world photomet- ric variations, a crucial feature for applications like cryo-ET analysis. While no single invariant model dominates across all domains, SEqSI presents the most compelling trade-off, delivering ex- cellent generalization to most OOD conditions while remaining competitive on in-distribution data.
7 EXPERIMENT 3: OBJECT LOCALIZATION
The task of object localization consists in estimating the position of gravity center of each object of interest in an image to either count them or study spatial distribution. It is common preliminary step instance segmentation methods: each location can be used to position a bounding-bov (Zhou et al. 2019 Duan et al., (019). For biological images, object localization has been largely studied with approaches locating object in an unsupervised manner in 2D (Midtvedt et al.. 2022). improving localization performances in case of very dense object distributions (Van De Looverbosch et al.
2025). A lot of segmentation methods designed for fluorescence microscopy (Schmidt et al. 2018 Weigert et al. 2020 Mandal & Chlmann 2021 Rapilly et al., 2025) first locate objects before initializing shape models. In microscopy imaging, photometric conditions can vary a lot, depending on microscope settings. Robust methods photometric corruptions
of particular interest to process image with unseen conditions. While some solution rely on training models on huge datasets (Pachitariu et al.) 2025), frugal approaches based on intrinsic robustness need be investigated, particularly in the case of 3D, where annotations are rare and costly.
7.1 EXPERIMENTAL SETUP
Architectures. We compare 4 U-Net architectures (Ronneberger et al.) 2015): Standard, SEq, Af- fEq and our SEgSI. The baseline networks (Standard and SEq) are trained using the approach based on sigmoid and BCE loss. Both AffEq and SEqSI are trained using the ZMSE and the standardized score maps (Eq. equation 2 ensuring theoretical invariance (see Supp. E.2. for details).
Datasets. We conduct experiments on two datasets: i) Data Science Bowl 2018 (DSB) (Goodman et al., 2018). a dataset of 2D real microscopy images from a Kaggle challenge, and focused on the 497 fluorescence images, each containing dozens of objects; ii) set of 3D synthetic images representing nuclei in fluorescence microscopy (see details in Supp. E.2.2).
Metric. Each method is evaluated via a score integrating between 0 and a maximum distance D, an accuracy curve (d + acc(d). The accuracy for given (in pixels) quantifies the amount of objects correctly predicted, meaning located at distance lower equal than d to their corresponding GT (see details in Supp. E.2.3 along with choice of D for each set).
Choice of the threshold value. For each method, the thresholding value used during the local maxima extraction, either after the sigmoid for classical approach or after standardization for our approach, is obtained by maximizing the score over the validation set.
We further show in App. D.2 that even with min-max normalization pre-processing, the Standard model fails generalize across domains, unlike our architecturally SEqST models.
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Standard (using normalization pre processsin))
SEq (using normalization as pre-processing)
Original image
Comppted Image
Original Image Score: 0.517
Corrupted image
Score: 0.573
Soore: 0.319
Score: 0.407
Inveriance
Invarience
AffEq
SEqSI
Original image Score: 0.502
Comuupted Image
Original Image Score: 0.643
Corrupted image
0.622
Invariance 0.202
Invariance measure: 0.719
Figure 4: Comparison behavior object localization between [0,1] normalization pre-processing and invariant approaches on spatially-varying shift. The image comes from DSB test set. The positions estimated on the clean (corrupted resp.) image in red (green, resp.). The shift here increases lincarly with the pixel position (from top left bottom right). The score compares the predictions with the GT positions. The invariance measure compares the predictions on the clean and corrupted image following the method of Supp. E.3.1 Non-visible red points correspond to perfectly overlapping green ones (invariance).
7.2 RESULTS AND DISCUSSION
Certified invariance. We conduct an empirical validation of the invariance of our approach on the 2D dataset of real images. Methods trained with input images normalized in [0.1] are evaluated for multiple scale, shift or affine transformations. We obtain similar results than those summarized in Table 2 only AffEq and SEqSI models with the standardized score-map guarantee affine invariance of locations prediction. All details on the invariance verification procedure are provided Supp. E.3. Performance results of models trained on the DSB set are illustrated in Fig.
Normalization pre-processing vs invariant models. While standard image normalization pre- processing (e.g., normalizing to [0.1] or perro-meanuuii--varinnee leads to global photometric affine invariance, it fails for spatially-varying ones. In contrast, our approach maintains robust perfor- mance even under spatially-varying corruptions. We compare the Standard and SEq (with [0,1] normalization pre-processing) and, AffEq and SEqSI under spatially-varying photometric shift. This experiment was conducted on the real world DSB set. The result for a shift, varying linearly in space, is illustrated in Fig. 4 Experimental details and theoretical invariance guarantees (focusing on the difference between normalization and the zero-sum layer of SEqS!) are provided App. Though weaker than for global case, the guarantees remain significant.
Binary segmentation. Binary segmentation is subtask of object localization, as score map thresholding is required for both. As segmentation masks are available for the DSB set, we evaluated our method on this set for binary segmentation. Theoretical guarantees hold (see Supp. F)
Robustness to Affine and Non-Affine Photometric Corruptions. Similarly to classification, we trained the models using different auteentumenttioon strategy and evaluated the location estimation on corrupted images (see experimental details in Supp. E.3.3). We see Table 5 that SEqSI is the best model for every augmentation strategy on uncorrupted images and all affine corruptions. Stand. trained with Aff. (or AlL. get close, but architectural design performs better than aug- mentation. For NAff. corruptions, SEqSI is less dominant, but with augmentation still perform very well on a few corruptions while others' performance drops. With augmentation it is the best or highly competitive. It is partieularly visible for the NAff. augmentation experiment: Aff. comuu-

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Table 5: Robustness to perturbations on the 3D synthetic test set of fluorescence microscopy. Test score for D = 6 of four architectures under photometric corruptions. Results grouped by the training augmentation strategy: none (0), affine (Aff.), non-affine (NAff.). or all combined (All). Each row corresponds to a different perturbation applied at evaluation. Within each training strategy (column group), the best result is highlighted gray. The overall best score for each perturbation is in bold. Results with std. given in Supp. Table 23
Train Aug.
Train Aug.
Alf
Train Aug. = NAHf
Train Ang. All
Corraption
Model
Stand
SEq
SEgSI
AdEEq
Stand.
SEq
SEgS1
Afti
Stand.
SEig
SE4S1
AllE
Stand.
SEq
SEgSI
Atlla
Original
0.868
0.605
112866
0.568
0.542
0.881
0.395
0.847
3.702
0:583
114833
0.877
0.853
0.377
tramsormations
0.152
0.117
012866
0.81
0.395
0.599
0.289
0.583
0.836
Stale
0.558
0.286
0.881
0.395
0.796
0.583
0.483
0.789
0,372
Stale
0.506
0.886
0.881
0.395
0.835
0.723
0.583
0.093
0.543
0.385
0.631
0.240
0.463
0.903
0.377
Non-Affine trandforaaionss
0.457
Scale saturated
0,823
Noiez low
0.505
0.385
Noise ligh
0.233
0.045
0.735
0.763
(dzrkee)
0.462
0,388
0.798
0.581
0.752
(ighten)
0.382
0.843
0.718
0.584
Additional experiment
0.571
0:842
0.587
0.816
0.509
D.477
0.277
0.530
0.845
0.352
0.576
0.777
0.521
0.787
1.472
0.789
0U064
0.0810
0.693
0.346
0.291
0.672
0.343
0.737
0.628
tions are addressed by design while NAff. corruptions were correctly learned using augmentation, making SEySI the best method on each corruption but one. AffEq, even though it provides iden- tical invariance guarantee as SE seems to be negatively affected by mata-augmentation, reducing its overall performances. Our model, on the other hand, continues to provide excellent results even with augmentation, which allows it to perform very well on non-affine corruptions too.
Robustness to bright artifacts. Artifacts are frequent perturbations in fluorescent microscopy, induced by sensor saturation (see supp. Fig. [T].. Those perturbations, which are neither global, nor affine, are nevertheless handled very well by our strategy, even without aata--uugmetaatiin (see last rows of Tab. 5). We attribute this to the fact that artifacts, together with the rescaling to the range [0, 11, locally act as a scaling effect away from the artifacts (details in Supp. E.3.4).
Conclusion. We provide method dedicated to object localization, applicable to 2D and 3D im- ages, that guarantees invariance to affine transformation. It performs very well on affine corruptions and, combined with data- augmentation, addresses wide spectrum of non-affine corruptions, mak- ing it a method of choice to perform object location estimation on non non-standardizer data.
CONCLUSION
In this work, we introduced the Ecale-Equivariant Shift-Invarintt (SEqSI) model, a novel architec- ture that provides practical path to photometric robustness. By prepending single shift-invariant layer to scale-equivariant backbone with appropriate post-processing. SE4ST achieves robustness to affine transformations by design, while remaining compatible with standard components like ReLU and incurring negligible computational cost. Our extensive benchmarks demonstrate that this ar- chitectural prior delivers significant practical benefits: SEqSI provides certified robustness to affine corruptions and enhances generalization to non-affine transformations. This robustness extends to challenging real-world domain shifts: SEqSI maintains high accuracy on OOD cryo-ET data where standard models collapse, and provides robustness to bright artifacts in microscopy object localiza- tion. This work establishes that architectural invariance, as implemented in SEqSI, is a powerful tool that complements data-driven techniques to build robust and high- seghh-eeffrmin models.
Future Work. Future work could explore SEqSI robustness to composite transformation groups like (e.g., MixUp (Zhang et al., 2018)) or data modalities like videos. Additionally, developing compatible normalization layers to further improve training stability and performance could be valuable direction (a preliminary work is available in Supp. (.7). Finally, we aim at extending the potential of our object-localization work segmentation.
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CODE AND DATA AVAILABILITY
All our implementations written in Python and are based on the PyTorch library. The code is
available on
com/ cww/MouniiiMioo.cco /wwuni//essauuui//// SSqqsi The datasets CIFAR-I0
(Krizhevsky et al. 2009) (Experiment 1), CZI cryo-ET challenge (Harrington al. 2024) (Experi- ment 2) and Data Science Bowl 2018 (Goodman et al. 2018) (Experiment 3) are publicly available. Access and processing details are given the README of the code and Appendix (see Repro- ducibility Statement 8). For the object localization experiment (Experiment 3), the 3D synthetic dataset will be made available upon publication. sample of the 3D synthetic dataset is given in Supplementary Materials.
REPRODUCIBILITY STATEMENT
We are committed to ensuring the reproducibility of our work. All implementation details for our models, including architectures, training procedures, and hyperparameter settings, are described throughout the paper and detailed in the Appendix. The full code and instructions to run all experi- ments are provided in the supplementary material. For clarity, we list the relevant appendix sections for each experiment below.
Experiment (CIFAR10 Classification):
Details on network architectures, training hyperparam- The photometric corruptions used for robustness A detailed protocol for invariance verification is available in
eters, and the CIFAR-10 dataset in Supp. C.I
evaluation are described in Supp. C.2
Supp. C.4
Experiment (Macromolecule Classification): Details on model architectures, training proce- dure, and data handling for the CZI Cryo-ET dataset (Harrington al. 2024) are detailed in Supp.
D.I
Experiment 3 (Object Localization): Architectural details for the 2D and 3D U-Nets are in Supp-
E.2.1 Specifics of the synthetic and DSB 2018 datasets Supp. E.2.2] The accuracy metric detailed in Supp. E.2.3] Settings for data augmentation and robustness tests are in Supp. C.2 and with further details on specific perturbations like bright artifacts in Supp. B.3.4
ACKNOWLEDGEMENTS
This work support by Agence Nationale de la Recherche to the HPC resources of IDRIS was granted under the allocation 2985521 made by GENCI.
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REFERENCES
Jérôme Boulanger, Alexandre Gidon, Charles Kervran, and Jean Salamero. A patch-based method for repetitive and transient event detection fluorescence imaging. PloS one, 5(10)e13190, 2010.
Dongdong Chen, Julián Tachella, and Mike E Davies. Robust equivariant imaging: a fully unsuper- vised framework for learning to image from noisy and partial measurements. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2022.
Taco Cohen and Max Welling, Group equivariant convolutional networks. In Maria Florina Balcan and Kilian Q. Weinberger (eds.), Proceedings of The 33rd International Conference on Machine Learning. volume 48 of Proceedings of Machine Learning Research, pp. 2990 2999, New York, mlr. press/
New York, USA, 20-22|1 Jun 2016. PMLR. URL btttps://proceediii
748/cohenc16.. ntml
Kaiwen Duan, Song Bai, Lingxi Xie, Honggang Qi, Qingming Huang, and Qi Tian. Centernet: Keypoint triplets for object detection. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), October 2019.
Allen Goodman, Anne Carpenter, Elizabeth Park, jlefman nvidia, Josette BoozAllen, Kyle, Maggie, Nilofer, Peter Sedivec, and Will Cukierski. 2018 data science howl. httts::/ com/ 2018. Kaggle.
competitions//eee
sance-bbwll.
Hao Guan and Mingxia Liu. A survey on domain adaptation for medical image analysis: concepts, methods, and applications. Journal of Applied Clinical Medical Physics, (3(1):e13472 2022.
Kyle Harrington, Mohammadreza Paraan*, Anchi Cheng, Utz Heinrich Ermel, Saugat Kandel, Dari Kimanius, Elizabeth Montabana, Ariana Peck, Jonathan Schwartz, Daniel Serwas, Han- nah Siems, Feng Wang, Yue Yu, Zhuowen Zhao, Shawn Zheng, Walter Reade, Maggie Demkin, Kristen Maitland, Dannielle McCarthy, Matthias Haury, David Agard, Clinton Potter, and Brid- get Carragher. Czii cryoet object identification. https:/ ptttpps://kggleee.. kag.alll....... wttpg//e..oo/ommeetttiacaoo compet itimstttiiins////
ion 2024. Kaggle.
Kaiming He, Xiangyu Zhang, Shaoging Ren, and Jian Sun. Deep residual learning for image recog- nition, 2015. UKL:/ttpps//arxivvoorr//b////
03385
Dan Hendryeks and Thomas Dietterich. Benchmarking neural network robustness to common cor- ruptions and perturbations, 2019. URL htttps:///// otg/abs/1903..ooomooooogo http:ppp.//aaaa.v..oo/o//////.13 12261
Dan Hendryeks, Norman Mu, Ekin D Cubuk, Barret Zoph, Justin Gilmer, and Balaji Thulasidasan. Augmix: A simple data processing method to improve robustness and uncertainty. In Interna- tional Conference on Learning Representations, 2020.
Sébastien Herbreteau, Emmanuel Moebel, and Charles Kervrann. Normalization-equiiaaaaa neural networks with application to image denoising. In Advances in Neural Information Processing Systems, volume 36, pp. 5706-5728, 2023.
Jonathan Krause, Michael Stark, Jia Deng, and Li Fei-Feii 3d object representations for fine-grained categorization. In 2013 IEEE International Conference on Computer Vision Workshops, pp. 554- 561. 2013. doi: 10.1 10111///......1.....
Alex Krizhevsky, Vinod Nair, and Geoffrey Hinton. Cifar-10 (canadian institute for advanced re- search). 2009. TtRLhtt::/// www. cs wtw.:caatttoooo ww..ootoo.duu.kriiaae.e wwfaariiaaaaaaaaaaeeeeeaaaaaaa.a cwf.a.itmalla.ll..............ll
Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convo- lutional neural networks. In Advances in Neural Information Processing Systems 25, pp. 1097- 1105, 2012.
Attila Lengyel, Ombretta Strafforello, Robert-Jan Bruintjes, Alexander Gielisse, and Jan van Gemert. Color equivariant convolutional networks. Advances in Neural Information Process- ing Systems, 36:29831-2988 2023.
12

---

Published as a conference paper at ICLR 2026
Soham Mandal and Virginie Uhlmann. Splinedist: Automated cell segmentation with spline curves. In 2021 IEEE 18th International Symposium on Biomedical Imaging (ISBI), pp. 1082-1086, 2021. doi:
Benjamin Midtvedt, Jesús Pineda, Fredrik Skärberg, Erik Olsén, Harshith Bachimanchi, Emelie Wesén, Elin K Esbjörner, Erik Selander, Fredrik Höök, Daniel Midtvedt, et al. Single-shot self- supervised object detection in microscopy. Nature communications, 13(1):7492, 2022.
Sreyas Mohan, Zahra Kadkhodaie, Eero Simoncelli, and Carlos Fernandez-Grand.. interpretable blind image denoising via bias-free convolutional neural networks. arXiv preprint (506::9900054777 2019.
Granda. Robust and
Marius Pachitariu, Michael Rariden, and Carsen Stringer. Cellpose-sam: superhuman generalization for cellular segmentation. bioRxiv, 2025. doi: 10.110122225.04.28.6660000 URL https://
www. htwp://wwww...rrrccoo.aee/aa////
04. 28. 24.60..5000.1.11111
Omkar M. Parkhi, Andrea Vedaldi, Andrew Zisserman, and C. V. Jawahar. Cats and dogs. In JEEE Conference on Computer Vision and Pattern Recognition, 2012.
Ariana Peck, Yue Yu, Jonathan Schwartz. Anchi Cheng, Utz Heinrich Ermel, Saugat Kandel, Dari Kimanius, Elizabeth Montabana, Daniel Serwas, Hannah Siems, Feng Wang, Zhuowen Zhao, Shawn Zheng, Matthias Haury, David Agard, Clinton Potter, Bridget Carragher, Kyle Harrington, and Mohammadreza Paraan. Annotating cryoet volumes: A machine learning challenge. BioRxiv, 2024. doi: 10. 10.1 10.11001/22241111 1.........66660 URL ltttps:////// www.bbiwwwwo..oooooo..... Barly/2024/11/66//
11 04. 621686
Quentin Rapilly, Anaïs Badoual, Pierre Maindron, Guenaelle Bouet, and Charles Kervrann. Pre- diction of parametric surfaces for multi-object segmentation in 3d biological imaging. In Ta- tiana A. Bubba, Romina Gaburro, Silvia Gazzola, Kostas Papafitsoros, Marcelo Pereyra, and Carola-Bibiane Schönlieb (eds.), Scale Space and Variational Methods in Computer Vision, pp. 255-268, Cham, 2025. Springer Nature Switzerland. ISBN 978-3-031-92366-11
Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedi- cal image segmentation. In International Conference Medical image computing and computer- assisted intervention, pp. 234-241. Springer, 2015.
Uwe Schmidt, Martin Weigert, Coleman Broaddus, and Gene Myers. Cell detection with star-convex polygons. In Medical Image Computing and Computer Assisted Intervention MICCAI 2018 2 Is International Conference, Granada, Spain, September 16-20, 2018, Proceedings. Part II, pp- 265-273, 2018. doi: 10.
Antonio Torralba and Alexei A Efros. Unbiased look at dataset bias. CVPR 2011, pp. 1521-1528. Ieee, 2011.
Tim Van De Looverbosch, Sarah De Beuckeleer, Frederik De Smet, Jan Sijbers, and Winnok H. De Vos. Proximity adjusted centroid mapping for accurate detection nuclei in dense 3d cell systems. Computers in Biology and Medicine, 185:109561, 2025. ISSN 0010-4825, doi: https:// boiilg//0..01/j...ombbmmmmo..d. 109561. htL www htttttpss/www..ceeeeeeeeeeeeeeee www.wciinceeiiee..t.... com/
Martin Weigert, Uwe Schmidt, Robert Haase, Ko Sugawara, and Gene Myers. Star-convex poly- hedra for 3d object detection and segmentation in microscopy. In The IEEE Winter Conference on Applications of Computer Vision (WACV), March 2020. doi: 10.1100/ACV44572.2200. 9093435.
Yulong Yang, Felix O'Mahony, and Christine Allen Blanchette Learning color equivariant repre- sentations. arXiv preprint arXiv:2406. (0.82406600988. 2024.
Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. In International Conference on Learning Representations, 2018.
Xingyi Zhou, Dequan Wang, and Philipp Krähenbühl. 1rX 2019.
Objects as points. arXiv preprint
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Appendix
Table of Contents
Large Language Models (LLMs) usage
15
B Methods Appendix
15
Mathematical foundations
15
B.2 Proof of Convolution and Linear Layers Equivariance and Invariance Properties
15
B.3
Proof Pooling Layers Equivariance
B.4
Proof MinMax--Normlizattion Invariance Affine Intensity Transformation
18
C
Experiments in Classification : CIFAR-10, Oxford-IIIT Pet and Stanford Cars Ap-
pendix
19
C.I
Details models implementation and data processing
19
Details on Photometric Augmentation and Evaluation Perturbation
22
C.3
Invariance by Post processing for Aremax based Tasks
23
4
Protacol for Invariance Verification
24
C,3
Numerical Stability of Weight Constraint Implementations
25
Full Robustness Results with Standard Deviations)
27
C.7
Compatibility Study with Normalization and Dropout Layers
28
C.8
Demonstrating Scalability More Complex Task: Oxford-IIIT Pets and Stan-
ford Cars Classification
C.9
Details 'Moderate' Photometrie Perturbations for Scalability Experiments
35
C.10 Transfert Learning from Standard model
38
D Experiments in Macromolecule Classification Appendix
40
D. Details on models implementation and data processing
40
D.2 Architectural Properties vs. Normalization preprocessing, for Domain Shift Ro-
bustness
44
Experiments in Object Localization Appendix
45
Details theoretical contributions
45
1.2
Details experimental setup)
47
E.3
Experimental results
50
F
Study of the binary segmentation task
59
F1 Details on experimental setup
59
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A
LARGE LANGUAGE MODELS (LLMS) USAGE
LLMs were used mainly to polish writing.
B
METHODS APPENDIX
B.1 MATHEMATICAL FOUNDATIONS
In this section, we denote T any group transformation.
Proposition 1 (Composition of Equivariant Functions) The composition of Tee-T---iiiiinee func- tions T eqquivariant.
Proof of Proposition 1 Let fi and f2 be two functions, equivariant to class of transformation T. For all T T. o Therefore, f1 f2 is also T-equivariant. The extension the composition of any finite number of equivariant functions is obtained by mathematical induction, proving the proposition.
f2
T fi f2 as both fi and f2 commute with any T.
Proposition 2 (Composition for Invariance) The composition of T Tequivariant function fol- lowed by -invariant function and then any function (with or without invariance or equivariance properties) is T -invariiant.
Proof of Proposition? 2
Let fi be equivariant T and invariant to T. Let fa be a function with no particular invariant or equivariant properties.
VT E T, fa f = fa o f2 o T fi by equivariance of f1,
fa f2 fi by invariance of ʃ2-
Therefore, f3 f2 fi is invariant to
In our case, we built the SEqSI network by positionning the iifft-invariatt layer at the very begin- ning (i.e. fi = La the identity function) without the use of any equivariant layer before.
B.2 PROOF CONVOLUTION AND LINEAR LAYERS EQUIVARIANCE INVARIANCE PROPERTIES
According to Herbreteau et al. 2023), ) let be linear or convolutional layer defined by its weights W and bias b. The action of the layer an input vector æ: is f(xx) Wæ + b. For convolutional layer, the operation is a convolution fxx = W * x + b, but the reasoning is still the same. An
affine transformation is defined as x (x) Ax #1, where 1 is vector of ones of appropriate dimension.
1. Scale-Equivariance (Tx.)) A layer f is scale-equivariant if f(T),(())) Tx)o((()) for any E Ri.
The left-hand side is: f(T2x(()) f(xx) W (Aæ) + X(W x) +
The right-hand side is: (f = Af(x)) A(W> + b) A( x) Ab.
For the equality to hold for any X, we must have Ab. Since this must be true for any À + 1, the only solution is b = 0. Thus, layer is cale-cquivariiant if and only if it has bias.
Shift-Equivariance (Ti.,) layer f is shift-equivariant if f(Ti,u())) = Ti,)(((x)) for any IR. Note that the shift value μ is preserved.
The left-hand side is: (x)) f(x +x+ = W +1) Wx p(W1) +b.
* The right-hand side is: Ti,, (f(x)) = f(x) + #1 (War x + b) + +l.
By identifying the terms, be true for any μ * O, this implies W1 1. This condition means that for each output neuron, the sum of its weights must be 1. Note that the bias term b does affect this property.
that the equality holds if and only if μ [[W 1) μ1. Since this must
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3. Affine-Equivriancce (Tx,u) A layer is affin--quuivariant if f(Tx, = (f)(f(z)) for
any (A, (1) E R x R. This requires combining the two previous properties. We set 0.
The left-hand side is: ((Txw())) f(x + +1) W( + μ1) \(Wx) + ((1))
The right-hand side is: (
Af(x)) + >(W=) jd.
The equality holds if and only if n (1) = ,1, which implies W1 1. Therefore, a layer is
affine-euuivariant if and only if its bias is null and its weights sum for each output neuron.
4. Shift-Invariance (Ti A layer is shift-invariant if ((T.,(())) f()) for any μ E R.
The left-hand side is: f(Ti,) (x)) = f(x + = W (1) Wx+ W +b.
The right-hand side is: f(x) = W æ + b.
The equality holds and only if (( = O. Since this must be true for any μ + O, this implies W1 = O. This condition means that for each output neuron, the sum of its weights must be O. As with shiff--quivvriiace,, the bias term does not affect this property. However, in architectures that also scale- equivvrqui (like SEqSI), the bias must be null.
5. Scale Invariance We did not identify any way to make a convolution scale-invariant while maintaining shift-invariance Applying log function could transform scale shift: log(Tx.o(z)) T (log(z)) and could be used with a shift-imvarian convolution. Unfortunately the log function is not compatible with shift as a negative shift would lead intput outside log def- convolution before the
inition domain. Considering first removing the shift using a log could be imagined but is not compatible with the required positivity of log input.
Padding Strategy. To preserve invariance/equivariance properties across the entire feature map, the padding strategy must be consistent with the corruption of the input image. Standard zero- padding breaks both shift-qquivariance and shift-invariance, it introduces fixed value (0) the borders that does not match according to the input signal. Instead, reflection padding must be used, it fills the boundaries with transformed versions of the input data, maintaining the integrity of the anuivariance/ivvaraaneee property. Other alternatives such as padding the image with fixed value that depend of the input image can be designed (e.g., padding with the min of the channel, or its mean).
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B.3 PROOF OF POOLING LAYERS EQUIVARIANCE
Let Spoot be pooling operation (max or average) applied over set of values X = f.1,..., xNN within local window. Let (x,(()) {Xæ; + y æ; X be the application of affine transformation with À E R_ and E R. The operation is Tar-equivaiaan if Spoot (T), Tx (foot (x)).
1. Max-Pooling.
The max-pooling operation is fman (X) , ØNJ..
The left-hand side is: fmaz (Tx,, (X)) max (xx) + p....., A + m...
Since À > O, the scaling is monotonic, and the addition of μ is a simple shift. Therefore, the maximum of the transformed values is the transformed maximum of the original values:
max Aax, 10) (a;
n..
The right-hand side is: Tr.a(mmun ()) = X(max (( + n.
Since both sides are equal, max-pooling is Tur-eruivariatt.
2. Average-Pooling.
The average-pooling operation is (X) 1 N
Ei= z4.
The left-hand side is: Sang (Tx, //2 Em (No, + +1).
By the linearity of the summation operator:
N
1 (1x +4) = N
Xax +
Σ
11
1
x + No
=
+H.
N
(5)
The right-hand side is: Tx (Sang ())
X
N
Σm
+ μ.
Since both sides are equal, average-pooling is also Tuuivarvari.
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B.4 PROOF OF MINMAX-NORMALIZTTUE INVARIANCE TO AFFINE INTENSITY TRANSFORMATION
We provide formal proof that normalization, such min-max scaling [0, 11, invariant global affine intensity transformations.
Let æ be an input image. An affine transformation Tx E Tatt applied, resulting in perturbed
image x' = x + p. with E IR and μ E R. The min-max normalization function is defined as:
min(@)
Norm ()) max (x) min(
(6)
We aim to prove that applying this normalization to the perturbed image x' yields the same result as applying to the original image æ, i.c., Norm (z') = Norm x)).
First, we determine the min and max of the perturbed image æ'. Due to the linearity of the transfor- mation and since À >
min(x') min(Aæ + p)) Ami (x) + f
max(=') max(Ax + μ1) À Amx((x) + μ
(8)
Now, we apply the normalization to w':
x' min(x:) Norm(x')= max(x") x') mminxx))
(Ax + (1) - (A (im((()) + +
(9)
(A max(()) + (Amin(x:) 14)
Simplifying the expression by canceling out μ in both the numerator and the denominator:
A mz)
X(x min(x))
Norm(a') mmxx (x) min(
(10)
Xmmax(ar) min(x:)
Finally, canceling the scaling factor
# min(x)
Norm
Norm
(1I)
max( min(
This proves that applying min max normalization as see-processing step after global affine inten- sity perturbation effectively cancels out the perturbation, making the input to the network invariant. A similar proof holds for z-score normalization.
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C EXPERIMENTS IN CLASSIFICATION CIFAR-10, OXFORD-IIIT PET AND STANFORD CARS APPENDIX
C.I DETAILS ON MODELS IMPLEMENTATION AND DATA PROCESSING
All classification models are based on ResNet-20 architecture He al. 2015 adapted for the x2x32 images of the CIFAR-10 dataset. The modular design allows us to instantiate the four model Comilies-Standdrte SEq. SEqSI, and -y applying the specific constraints summarized in Table of the main paper. A key difference lies in the padding strategy as described in Appendix B.2 Standard and SEq models use zero-pas aro-ppdding, whereas AffEq and SEqSI models use reflection padding to preserve their respective equivariance and invariance properties.
Overall Architecture. The network follows a ResNet-20 structure adapted for 32x32 images, comprising an initial convolutional layer, three residual stages, and a classification head.
The initial layer is a 3x3 convolution mapping the 3-channel input to 64 feature maps while pre- serving the x2x32 resolution. Its properties are configured for each model family as per Table for instance, it is shift-invariant (weights sum to 0) for SEqSI and affin-equiivaiant (weights sum to 1, bias-free) for AffEq.
The network then proceeds through three residual stages, each with 3 residual blocks. Stage operates on 64-channel feature maps. Stages and 3 double the channel count to 128 and 256, respectively, while halving the spatial resolution to 16x 16 and 8x8 8 pixels. This downsampling is performed by the first block of each of these stages, which uses a stride 2 in its first convolutional layer.
The classification head processes the final 8x8x256 feature map. global average pooling layer reduces it to 256-dimensional vector, which is then projected onto 10 output logits by fully connected layer. final 'softmax' function converts logits to class probabilities.
Residual Block Structure. Each of the 9 residual blocks consists of a main path and a shortcut connection.
The main path contains two 3x3 convolutional layers. Based on the implementation, the data flow Cone + Norm -> Activalion - Dropout Conv Norm - Dropout. The output of this path is then added to the output of th shorteut, and a final activation is applied to the sum. The specific activation ('ReLU" or 'SortPool') depends on the model family (T.
The shortcut connection is an identity mapping if input and output dimensions are identical. In downsampling blocks, it employs 1x1 convolution with stride of match the output dimen- sions of the main path before the element-wise addition.
Optional Layers for Compatibility Study. By default, none of the model families include nor- malization dropout layers, ensuring a baseline comparison focused purely on the architectural constraints. For the compatibility study presented in Section C.7] these layers are optionally in- serted into the residual blocks. Normalization layers (InstanceNorm, BatchNorm) are positioned after each convolutional layer but before the activation function. The Dropout layer is inserted after the activation function.
Data Preprocessing and Data Augmentation. To ensure fair comparison and enlighten the benefits of architectural invariance, all models are trained with carefully designed data pipeline. A key aspect of this pipeline is the exclusive use of geometrie data augmentation. We decided to avoid photometric augmentations (c.g., brightness contrast changes) to prevent models from learning invariant properties. This setup guarantees that the observed robustness to photometric corruptions is direct result of architectural design, not from the training dataset.
The pipeline begins by scaling input images values from the original [0, 255] range to [0, 1] (a simple division by 255 is applied). During training, then apply the following geometric augmentations to enhance generalization:
Random Horizontal Flips: Each image is flipped horizontally with probability of 50%.
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Random Cropping:
The 32x32 imput images are first padded with 4 pixels each side create 40x40 image. A random 32x32 crop is then extracted from this padded image.
For model selection, evaluated performance augmentation; only the scaling to [0, I] range applied. This standard procedure ensures a fair comparison, the best checkpoint for each model is selected based on its performance the clean, canonical data distribution, prior to the final robustness evaluation on the test set.
the validation set without applying any data
Training Details. To ensure fair comparison, all models were trained under identical conditions. We train each model for 500 epochs with batch size of 128, using the standard Cross-Entropy loss. Optimization was performed with SGD, with momentum of 0.9 and weight decay of 5e-4. The learning rate was initialized to 0.1 and followed cosine annealing schedule. We applied gradient clipping with maximum norm of 1.0. To ensure robust results, repeated each experiment with distinct random seeds (5, 7, 42, 137, 181). For each run, we selected the checkpoint with the highest validation accuracy and reported the mean and standard deviation of the test performance between seeds. We use the standard CIFAR-10 splits, reserving 10% (5000 images) of the training for validation. All experiments were performed on a single NVIDIA A40 GPU, The training and validation curves in Supp. Figure 5 shows that Standard, SEq and SEqSI converge similarly, while AffEq takes more time to converge.
80
60
40
Step
20
0
100
200
300
400
AffEq wartq_val__ae val_accuaacyyyy
Stand_ traitrannnaaaat accurrcy
-- Stand_ _lal_llllaa_ sal_accuracy
SEq train accuracy -- SEq val. accuracy
SEqSI train accuracy
-- SEq val accuracy
(a) Training and Validation accuracy
2
1.5
0.5
Step
100
200
AffEq train loss
AflE val. loss
Stand traint loss
Stand loss
SEq train.nn loss
SEq val loss
SEqSL loss SEgSLvvd
(b) Training and Validation
Figure 5: Training dynamics CIFAR-10. Validation accuracy (a) and loss (b) curves for all ResNet-20 variants.
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Table 6: Computational costs for each model, measured an NVIDIA A40 GPU. Metrics are averaged over 495 epochs after a 5-epoch warmup. Training (s/epoch): Time training epoch on 45,000 images. Inference (s): Total time for the 5, 5,000-image validation Peak Mem. (GB): Maximum GPU memory allocated during training. # Parameters: Number of trainable parameters.
Model
Training (s/epoch)
Inference (s)
Peak Mem. (GB)
# Parameters
Standard
11.87
0.66
0.44
4122
SEq
10.79
0.61
0.44
4321 472
SEqSI
11.23
0.61
0.44
4321 472
AflEq
18.61
0.86
1.26
4321 481
Computational Costs. Supp. Table detaalls the computational costs for each tested model. The
results highlight the efficiency of the SEqSI model, which operates with a computational footprint nearly identical to the SEq and even better than Standard. In stark contrast, the AffEq model is substantially more resource-intensive with training time over 50% longer, an inference time over 30% longer, and a peak memory usage nearly three times higher.
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C.2 DETAILS ON PHOTOMETRIC AUGMENTATION AND EVALUATION PERTURBATION
The photometric transformations used for both data augmentation and evaluation are detailed below. Note that the parameters are the same for transformation and augmentation, respectively, inference and training time. The parameters correspond to those described in Figure To ensure experimental consistency, the same parameters are used for Experiment 3: Object Localization (Section 7).
Augmentation Pipeline. During training, for the 'Aff", 'NAff', and 'All' strategies, we applied photometric augmentation using 'transforms. Raniomss.RnddoChoicee mechanism. For each image, one transformation randomly selected from the corresponding set and applied. This set also includes an identity transformation, meaning there a chance an image is not photometrically augmented. The parameters for each transformation (e.g., the specific shift value) sampled uniformly from the specified range for each image independently. All photometric augmentations are applied after geometric augmentations (random crops and flips).
Evaluation Protocol. For the robustness evaluation (Table each transformation type was tested given corruption (e.g.. 'Shift'), we applied the corre-
independently. To measure robustness sponding transformation to each image in the test set. Crucially, and similarly to the augmentation pipeline, the specific parameter for the transformation (e.g., the value of ju for shift) was sampled independently and uniformly from the specified range for each image. This ensures that the evalu- ation assesses performance across the entire distribution of the corruption, rather than just a single fixed value.
Transformation Groups. The transformations are categorized into two main groups:
Affine Transformations (Aff): This group consists of 4 non-saturated global affine transformations. They form the 'Aff' augmentation set.
Global Additive Shift:
T(x) = x + μu, with μ ~ u[-2.0,
Global Scaling (Compression): T(x) Ax, with 4[0.0,
Global Scaling (Dilation): T(x) Aa; with À 41110, 4.0).
Global Affine: T(x) A + μ., with A ~ (0.0, 4.0) and μ ~ 210 2.0].
Non-Afline and Saturated Transformations (NAff): This group includes 9 transformations that are either non-affine, spatially-varying, saturated. A saturated transformation means the output pixel values are clipped to the [o. range. This group forms the 'NAff' augmentation set.
Saturated Additive Shift: T(x) = clip(= + μ4 O, 1), with ~ 0[-0.7,0.7]
Saturated Scaling: T(x) clip A2x 1), with
3.00.
Saturated Affine: T'(x) = clip 0, 1), with X ~ 24 1.3] and p 0[-0.3,0.3)
Spatially-Varyine Affine: T(x)(u,,) v) X(u, v) v) p(u,)), v), with X(u,v) ([(0,1, 0.5] and u(u, ~ 1] for each pixel (u, v).
Contrast Inversion:
spatially-varyy varying scaling T(x)(u, (),,v)=(u, u). with (u,,))~Z(([ -022 for each pixel (u, v).
Additive Gaussian Noise (Low): T(x) = clip(# + 0, 1), where n ~ oN(0,²²) and
a ~ 20000000003
Additive Gaussian Noise (High): T(x:) clip(= n, 0, 1), where o2 and a ~ 03/(0.15,0022
Gamma Correction (Lighten): T(x) = x², with y ~ (4.0221.0)
Gamma Correction (Darken): T(x) x², with 7 4[1.0, 5.0).
The 'All' augmentation strategy combines both 'Aff' and 'NAff sets, for total of 13 transforma-
tions.
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C.3 INVARIANCE BY POST-PROCESSING FOR ARGMAX-BASER TASKS
This section proves that for any task where the final prediction is derived from an 'argmax' operation (e.g., classification, semantic segmentation), the prediction invariant to input affine transforma- tions when using either an AffEq or SEqSI model.
Let f()) be the logits produced by a network for an input æ. When the input is transformed by T (a) = Aæ + pl, the logits x' transform differently depending on the architecture:
For an Affine-Equivariine (AffEq) network: z' = f
( (x)) = Az + 1ll
For a Scale-Equivariant, shift-invariant (SEqST) network: = f(T), Az.
Both cases can be unified by considering the general transformation z' = A= μ1l, where μ 0 for the SEqSI model. The core of the proof that the 'argmax' operation is insensitive to any strictly monotonic increasing transformation of its input vector.
1. Invariance of Argmax. The transformation + z' A= + μ1 is strictly order-preserving for any À > That is, 2; > 2j for any two elements of 2, then their transformed values also satisfy λæ; + f > Xzj + Consequently, the transformation does not change the index of the maximum element:
arg max(=') arg max (Az + m1) = arg max(=).
(12)
This shows that applying 'argmax' directly to the logits of either an AffEq or SEqSI network yields a provably invariant prediction.
2. Invariance with Strietly Monotonic Increasing Activation. Now, consider applying strictly monotonic increasing function to the logits before the 'argmax', like 'softmax'. Since h is strictly monotonic increasing, it also preserves the order of its inputs. Therefore, the order of elements in (22) is the same as in z, and the order in h(z') is the same as z'.
arg max (h(2²)) = arg max(=') = arg max (2)x arg max (h())..
(13)
This demonstrates that the composition 'argmax' 'h' is invariant to affine transformations of the logits, provided h is strictly monotonic increasing. This justifies its use as a final post-processing step for any argmax-based task, guaranteeing that the final prediction is robust affine photometric transformations of the input image.
Practical Consideration for Numerical Stability. In practice, when dealing with very large in- put transformations, the logits z can reach extreme values, potentially causing numerical over- flow activation functions like 'softmax'. the logits by subtracting their maximum value before applying the activation, i.c,, computing h( max(=)). This operation does not affect the final prediction, as 'argmax" is invariant to this shift: arg max(h(z max(2))) arg max(= max(z)) arg max(2). This makes the computation robust to large shifts in logit values.
standard technique to ensure stability to shift
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Table 7: Mean and standard deviation of prediction invariance error (%, lower is better) over multiple seeds, and different degrees of affine transformation. Format is mean + std. Best mean result per row in extbfbold. Note that, 'dil.' and 'comp.' stand for dilation and compression, respectively.
Transf.
Setting
Standard
SEq
SEqSI
AffEq
(left shift, moderate)
90.0 +0.37
89.4 +0.59
0+0.00
++0.00
= 0.5 (right shift, small)
12.2 +0.78
11.4 +1.21
+00000
0 +0000
Shift
2 (right shift, moderate)
58.1 +2.71
58.6 +.225
+00.00
0 +0.00
10 (right shift, extreme)
88.0 12.39
87.4 12.58
1:0.00
0 10.00
0.5 (comp.)
8.48 +0.64
0 +0.00
#=0000
O =0.00
Scale
3.0 (dil., moderate)
4.48 10.47
0 10.00
10.000
0 10.00
255 (dil., extreme)
5.93 10.70
1.0.000
100.00
10.000
-2, 10 (left shift strong dil.)
8.05 +0.35
6.78 +0.20
#0.00
=0.00
Affine
F À
0.1 (shift + strong comp.)
90.2 +:026
89.5 +0.59
1:0.00
100000
5, À 3 (shift + dilation, moderate)
49.3 4:2.99
49.5 +3.93
#00.00
0 100.00
C.4 PROTOCOL FOR INVARIANCE VERIFICATION
To empirically verify the theoretical invariance properties of our proposed architectures, we con- ducted an evaluation on the best performing checkpoints, selected based on validation accuracy after training. This step is crucial to ensure that the assessment of invariance is not confounded by arbitrary behavior of an untrained model, where initial weight values could themselves influence predictions. For each model, trained with one of our 5 distinct seeds, applied set of affine transformations (shift, scale, or both) to the entire CIFAR-10 test set. While this evaluation is de- terministie, set the random seed to match the used for training each model to ensure full reproducibility.
Pred. Inv. Err.
N Σ (argmax(f(Tar(x))) argmax(f(x:))) 100
(14)
where I is the indicator function and N is the number of images.
We then measured the prediction invariance error (see Supp. Equation 14 defined as the per- centage of images for which the predicted class changes once the transformation has been applied. To accurately measure perfect invariance (0% error), this test is run in float64 precision, as stan- dard float32 can introduce numerical errors that break theoretical guarantees (see Appendix C.5 for more details). This protocol allowed to enlighten us to isolate and quantify the inherent robustness provided by each architectural design.
In Supp. Table 7 we reported the prediction invariance error for each model family. To test these properties, we applied range of affine transformations (shift, scale, and their combination) for varying intensities, from moderate to extreme. This approach enabled to quantify how the invariance error changes with the magnitude of perturbation, highlighting the stable, perfect invariance of the certified models in all conditions.
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Table 8: Prediction invariance error (% mean - std) in float32 precision. Comparison of
"Telescopic" and "Mean Subtraction" implementations. Values of 0.00 are highlighted in gray.
SEqSI
AffEq
Transform.
Setting
Telescopic
Mean Subtr.
Telescopic
Mean Subtr.
μ
0.01 10.00
0.00 10.00
0.04 10.01
0.05 10.02
μ = 0.5
0.00 10.00
0.00 10.00
0.03 10.01
0.02 10.01
Shift
l = 2
0.01 10.01
0.00 10.00
0.04 10.02
0.06 10.02
μ = 10
0.00 #0.01
0.00 #0.00
0.22 #0.03
0.26 =0.05
À = 0.5
0.00 +0.00
0.00
10.00
0.00 10.00
0.00 1:0000
Scale
À = 3
0.01 +0.00
0.01
#0.01
0.02 +0.01
0.02 +.001
À 255
0.01 #0.01
0.00
#0.01
0.02 #0.01
0.02 #0.01
J 22, A 10
0.01 +0.01
0.00 +0.00
0.02 +0.01
0.02 +0000
Affine
Jr 5, 0.1
0.01 +0.01
0.01 +0.00
1.16 +0.12
1.41 +0.10
# = 5,
0.01 10.01
0.01 #0.01
0.04 10.01
0.03 10.01
Table 9: Prediction invariance error (% as mean + std) in float64 precision. Comparison of "Telescopic" and "Mean Subtraction" implementations. Values of 0.00 are highlighted in gray.
SEqSI
AffEq
Transform.
Setting
Telescopic
Mean Subtr.
Telescopic
Mean Subtr.
# -2
0.00 10.00
0.00 #0.00
0.00 #0.00
0.00 #0.00
μ = 0.5
0.00 10.00
0.00 #0.00
0.00 +0.00
0.00 =0.00
Shift
μ = 2
0.00 +0.00
0.00 #0.00
0.00 +0.00
0.00 =0.00
μ = 10
0.00 +0.00
0.00 +0.00
0.00 +0.00
0.00 +00.00
À = 0.5
0.00 #0.00
0.00 #0.00
0.00 #0.00
0.00 =0.00
Scale
À 3
0.00 10.00
0.00 10.00
0.00 10.00
0.00 10.00
À 255
0.00 +0.00
0.00
+0.00
0.00
+0.00
0.00 +0.00
μ À 10
0.00 10.00
0.00
10.00
0.00 10.00
0.00 10.00
Affine
# 5, 0.1
0.00 10.00
0.00 10000
10.00
0.00 10.00
A = 5,
0.00 +0.00
0.00 =0.00
0.00 +0.00
0.00 +0.00
C.5
NUMERICAL STABILITY OF WEIGITT CONSTRAINT IMPLEMENTATIONS
The weight constraints for the AffEq and SEqSI models (respectively, Σ w = 1 and Σ w = 0) can be enforced in several ways. We compare common parameterization methods: "Telescopic" and "Mean Subtraction".
w' roll(w) w
(Telescopic)
SEqSI (shift-inv., Σni w' 0):
w = w
N "
wi
(Mean Subtraction)
(15)
1w' roll(w)
(Telescopic)
AffEq (shift-equiv. Σ w/' = Iw'=1):
1):
N
w' = w Ee
w. +
(Mean Subtraction)
N
where (w) denotes a circular shift of the vector w by one position (i.e.. roll(w): w;-1).
Both the "Telescopic" and "Mean Subtraction" implementations are theoretically invariant. How- ever, we observed that numerical approximations in float32 precision can break these theoretical guarantees. As shown in Supp. Table the key finding is that numerical stability is mainly driven by the architecture (SEqSI vs. AffEq) rather than the specific parameterization. While both imple- mentations for SE are highly stable with only negligible errors, the AffEq model is significantly more prone to instability, exhibiting much larger errors regardless the parameterization. Within the SEgSI model, the "Mean Subtraction" variant is marginally more robust. As detailed in Supp. Table 9 perfect invariance was restored for all models and implementations when performing cal- culations in float64 precision.
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The "Mean Subtraction" method was used for all classification-tase experiments. Invariance exper- iments were performed in float64, while performance experiments were performed in float32 for computational reasons. The object location experiments were conducted with the "Telescopic" im- plementation float64 precision. We reported here implementation details for reproducibility of results.
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C.6 FULL ROBUSTNESS RESULTS WITH STANDARD DEVIATIONS
In Supp. Table T0 we present the full robustness results, corresponding to the main results in Table reported the mean and standard deviation of accuracy over
To ensure statistical significance, complete training and evaluation runs, each performed with a distinct random seed. The results are split into two parts for readability. Note that all experiments were conducted in float32 precision.
Table 10: Robustness to evaluation-tims perturbations (mean + std over seeds). Test accuracy (%) of four architectures under various photometric corruptions. Models are grouped by the training augmentation strategy used: none (Ø), affine (Aff.), non-affine (NAff.), or all combined (All). Each row corresponds to different perturbation applied at evaluation. Within each strategy group, the best result highlighted in gray. The overall best result for each perturbation in bold.
Train Aug. Ø
Train Aug. AfT
Model
Corruption
Stand.
SFg
SFgS1
Afflig
Stand.
8Fq
SEgg11
AffFig
Original
91.7 10.3
91.3 #0.1
91.2 20.3
89.6 #0.1
91.2 #0.3
90.6 +:01
91.0 #0.1
88.9 4002
Affine transformations (AfL)
Shift
53.4 +1.2
91.2
40.2
91.0
+0.2
Scale (< 1)
68.0 10.8
91.3 10.1
91.2 20.3
89.6 +0.1
90.2 +0.5
90.6 4:011
91.0 20.1
88.9 4:0.
Scale (>>
91.0 10.1
91.3 10.1
91.2 40.3
89.6 10.1
91.1 10.3
90.6 10.2
91.0 #0.1
88,9 +0.2
Affine
65.1
89.6 +:.1
90.7 :0.33
91.0
88.9 1:0.2
Non-Affine transformations (NAfL)
Shift saturated
72.0 :10.6
73.4 1:0.4
79.0 2:066
76.7
10.4
79.3 10.6
79.0
1:0.8
78.4 #0.6
76.1 +:0.3
Scale (>> 1) saturated
78.1 10.5
78.3 10.6
77.6 20.5
76.0 +0.5
78.0 10.7
77.7 10.7
76.9 +0.5
75.7 40.3
Affine saturated
59.9 1:.88
39.6 11.1
12.3 110
70.9 10.5
72.610..
73.3 11.0
71.7 20.7
70.6 4:0.3
Spatially-varying Affine
31.8 +1.0
38.2 +2.3
72.5 +2.2
63.6 1100
90.2 20.6
88.3 10.4
73.0111
66.6
Contrast Inversion
9.6
15.4
52.9 +1.0
61.3
52.9 +1.3
Noise (low)
90.8 40022
90.6 +0.2
90.1 20.6
88.5 +0.2
90.0
89.8 +002
90.0 =0.1
87.8 +0.2
Noise (high)
19.0422.4
22.4 +3.5
15.2 +1.0
20.3 +0.6
15.5 +1.11
19.8 4-3.4
16.0 4-22
19.2 +1.9
Gamma (darken)
11.0
76.4
76.4
82.5 10.3
11.1
10.4
76,2 40.2
Gamma (lighten)
85.9 10.6
85.3 +0.7
90.3 20.2
88.6 10.1
90.6 #0.4
89.9 10.1
90.0 #0.2
88.3 +0.2
Train Aug. NAIT
Train Aug. All
Model
Corruption
Stand.
SFg
SEgS1
Affli
Stand.
SFg
SFg81
AffFig
Original
92.1 +0.1
92.1 +0.2
91.5 +0.2
90.5 +0.1
92.0 40.1
91.6 +0.1
91.8 +0.2
90.6 +-022
Affine transformations (Aff.)
Shift
88.5 +0066
88.7 +0.3
91.5 40.2
90.5 +0.1
91.4 +0.2
90.7 +0.2
91.8 40.2
90.6 +-0.2
Scale (<
92.1 10.2
10.2
90.5
90.6 +:0.3
91.8
90.6 4:02
Scale (> 1)
91.6 +0.1
92.1 +0.2
91.5 +0.2
90.5 +-0.1
92.0 +0.2
91.6 +0.11
91.8 +00.2
90.6 +-0.2
Affine
87.0 10.88
88.1 10.4
91.5
20.2
90.5 1:011
90.9 ::0.2
90.1 10.11
91.8 20022
90.5 4:0.2
Non-Affine transformations (NAfL)
Shift saturated
87.3 10.2
86,9 1:0.2
86.4 +0.33
85.0 +:03
87.1
::0.5
86.1
1:044
86.4 10.11
84.4 1:0.5
Scale (> 1) saturated
88.0 +0.11
87.5 +0.3
87.1
+0.3
85.6 +0.33
87.9 +0.2
86.7 +0.1
86.8 +0.2
85.0 +-0.3
Affine saturated
84.8 10.3
10.3
82.8 10.3
84.9 10.3
83.7 10.3
81.9 10.5
Sputially-var Affine
88.2 #0.6
87.0 +0.2
89.6
20.2
88.6 +0.2
89.9 +0.2
89.3 +0.2
89.4 #0.1
88.1 +0.3
Contrast Inversion
90.2 +0..2
90.3 +0.2
+0.2 87.6
86.3 +0.4
89.2 40.44
87.8 +0.2
86.5 +0.1
84.6 40.4
Noise (low)
#0.1
91.7 +0.2
91.0
90.0
91.5 +0.2
91.3 #0.4
+0.2
Noise (high)
79.6 +0.3
80.0 +-0.6
78.7
+0.7
76.6 +0.5
78.3 +0.4
78.3 +0.4
77.9 +0.5
75.2 +0.9
Gamma (darken)
89.6 10.2
88.8 10.4
88.3
10.2
86.6 10.3
89.5 10.4
87.8 10.1
88.0 10.2
86.0 46.0
Gamma (lighten)
91.8 +0.2
91.9 000.2
91.1 40.2
90.2 +0.1
91.7 +.22
91.3 +0.1
91.5 40.3
90.2
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Published as a conference paper at ICLR 2026
C.7 COMPATIBILITY STUDY WITH NORMALIZATION AND DROPOUT LAYERS
Analysis of Compatibility We conducted a study to assess the compatibility of our proposed ar- chitectures with standard deep learning components that are known to improve performance and training stability: normalization layers and dropout. For this study, we integrated Batch Normaliza- tion (BN), Layer Normalization (LN), Instance Normalization (IN), and Dropout (DP) into each of the four model families. All layers were used with their default PyTorch parameters to reflect com- mon practices. We also evaluated a "Mix" strategy, which employs Instance Normalization after the first convolutional block and Batch Normalization in all subsequent blocks, a technique intended to combine the benefits of both. The results, presented in Supp. Tables I and 12| are analyzed in the following from the perspectives of both theoretical invariance and empirical performance.
Impact on Invariance Guarantees. The invariance error mcasurements (Supp. Table reveal how these layers interact with the architectural constraints.
SEqSI Model: A key finding is the robustness of shift-invariance. This property, enforced by the first layer, is perfectly preserved (0% error) across all configurations, including all normalization types and dropout. However, its scale-equivariacce is broken by all normal- ization layers (BN, LN, IN, and Mix).
The more restrictive AffEq model is less compatible. Its affine-equivariance is completely broken by Batch Normalization and partially by Layer Normalization. It only maintains its guarantees with Instance Normalization, which itself saffine-ivvariant (where is value added to the denominator for numerical stability. Default: le-5.)
AffEq Model:
Standard Components:
Dropout, being inactive at inference, does not affect the invari- ance of any model. Instance Normalization, because of its own invariance properties, can provide shift-invariance to the Standard and SEq models, but at the cost of breaking the scale-equivariance needed for a principled design like SEgSI
This analysis confirms that integrating standard normalization layers breaks the theoretical scale- equivariance of the SEgSI model. The magnitude of this impact on scale-invariance varies sig- nificantly: Batch Normalization severely degrades it (e.g., up to 86.8% error), whereas Layer and Instance Normalization have a much smaller impact, with an error typically below 1% but reaching up to 4.93% in some cases. Crucially, the shift- inft-invariiance guarantee remains perfectly intact across all normalization layers. This establishes a practical trade-off, which we now evaluate by analyzing the impact performance.
While normalization layers can break theoretical invariance properties, the performance is still preserved in practical experiments (see Supp. Table 12p
Impact on Performance.
Without normalization, the architectural constraints of SEqSI re- sult in a clean accuracy of 91.2%, slightly below the Standard model accuracy (91.7%). However, this gap is reduced and often reversed when normalization is introduced. SEqSI benefits immensely from normalization, with its performance increasing by over 4 percent- age points to 95.7 7% with BN and 95.5% with the Mix strategy. This synergy elevates SE to match or exceed the performance of the Standard model on clean data, demonstrating its compatibility with modern training recipes.
Synergy with SEqSI:
Enhanced Robustness:
The combination of the SEqSI prior and normalization layers leads to remarkable empirical robustness, even when thcoretical scale-equivariance is partially broken. When paired with normalization, SEqSI consistently outperforms its Standard counterpart. For instance, the SEgSI+Mix variant is superior to Standard+ +ix on 8 out of 13 corruptions, with particularly large gains challenging non-affine transformations like "Spatially-varying Affine" (94.9% vs. 89.4%) and "Contrast Inversion" (71.7% vs. 66.7% (%.% This demonstrates that the architectural properties provides powerful inductive bias for learning robust features, which is further amplified by the stabilized training dynamics from normalization.
Practicality over AffEq: In contrast, the AffEq model, while theoretically well- -rolnggo proves less in practical imaging. Its performance on clean data drops significantly with Dropout (from 89.6% to 81.2%), and it is consistently outperformed by SEggtt across all
28

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Published as a conference paper at ICLR 2026
normalization schemes. This underscores SEqSI superior balance of theoretical principles and practical utility.
This study demonstrates that while SEqSI compatible with standard normalization layers at the cost of its theoretical scale-equivariance, the practical benefits are substantial. The combination of SEqSI architectural properties with normalization layers results in models that not only match the performance of standard networks on clean data, but consistently outperform them in terms of robustness to a wide range of photometric corruptions. This synergy establishes SEqSI as a practical, effective, and principled choice for building high-performing and robust networks, making it a valuable tool for real-world applications where data variability is a key challenge.
Conclusion:
Table 11: Detailed invariance error (% as mean + std over seeds) for different normalization layers.
The best result (lowest error) per row is in bold. A value of 0 indicates perfect invariance and is highlighted in gray. Ø: no Norm, BN: Batch Norm, LN: Layer Norm, IN: Instance Norm, Mix: Mixed Norms, DP: Dropout.
DP
Transform.
Setting
Stand
SEq
SE45I
AffEq
Stand.
SEq
SEx51
AffEq
- (Ielt shift, medeerate)
90.0 +0.37
89.4 +0.59
0000
#5000
0.00
20:00
89.3 +00.7
80.1 #:1.70
0.00
#0:000
0.00 2:0000
Shift
U.S crigle shift, small)
fright shift, modernse)
56.1 +2:71
12.2 +0.78
58.6 13.25
+1:21
0.09 10000
0.00 #3000
0.00
0.09
2:.00
10:00
12.3 +..50
55.7 4:2.6
8.27
51.3 1:160
2:3.38
0.00 1:0000
0.00
2:0.00
0.00 1:0:00
0.00
2:1000
10 (right shift, extrame)
88.0 +2.30
87.4 +2.58
0.00 +0.00
0.00
+0.00
88.0 +1195
88.8
+0.94
0.00 +-000
0.00 +:0.0
- (compresiion)
8,88 10.64
0.00
30.0
0.09
S50000
0,00
35000
7.31 3:100
0.00
S23100
0.09
2200000
0.00
200100
Scale
- 3.0 callatio, modense)
4,48 1.0.47
0.00
10.00
0.09 150000
0.00
3.000
10.6 1.0.9
0.00
211000
0.00 1:0000
0.00 1:1100
255 (dilation, extreme)
5.99
+0.70
0.00
+0.00
0.00
+-000
0.00
+0.00
30.5 +.100
0.00 +0300
0.00
+-0.00
0.00
+-0.00
2.A 10 fleft stift e strog dilasion)
8,05 1:0.35
6.78 40.20
0.00 350000
0.00
350001
19.0 3:1.71
5.82 2:1224
0.09 1:0000
0,00 30000
Afline
5. 0.1 (sbiit e strong compression)
90.2 1.0.26
10.39
0.09 1,000
0.00
1..00
99.5 1.0.1
39.7
1.111
0.00 1:0000
0.00
1.1000
5,A (sttftt + dilrioo, moderate)
49.3
1.2.99
49.3 13.93
0.00 10.00
10.00
61.7 16600
41.4 1..52
0.00 1.000
0.00
10.00
BN
LN
Transform.
Setting
Stadd
SBiq
SDgS1
Atlla
Stand.
S0g
S0qS1
Atllq
modernt)
00.9
10.09
1-0.00
P. (right whift, smmlll)
00.00
12:2-+1.:00
7.65
P lright mederate)
27.4
A 10 (right shift, extreme)
20.02
89.9 1:005
0.00 1:000
90.0.100:00
62.2 5..81
13.56
0.00 1:000
1.000
3 (cmmpecssion)
4.00-0.32
4.06 +:3.10
5:91
#0.03
+0000
3 (dilation, modecate)
56.7
#0.01
8:000
À 255 (dilation, extemmo
89.9:1.30
90.3 8:
86.8 2:277
90.1
7.24 11.19
0.01
30.01
0.10
10000
1:0.1
F (shift stron compecssion)
shilt strong dilation)
6.63
#0.62
#
72.3
20.41
# 5. (shift dilation, modeenee)
90.0-0.18
+0
28.3
5.999
Mix
Transform.
Setting
Stand
SEgSI
AffEq
Stand
S6qS1
AffTq
(left stif, mndentt))
6:66 =0.63
5.80 #0.31
0000 2:000
0J00
200.00
5.48 =0.25
3.22 +0.09
10,0 =60000
3.00-20000
Shan
0.5 (right shift, small)
shift, mederate)
903 903-0.37
3.00:.:20
8.61 10.10
3.26 10.05
0.00 10:00
0.00 10000
0.00 10.00
0.00
1.81 -0.11
5.10-0255
4.91 10.56
1:83 10.11
0.00 10000
0.00
1:000
0.00 1:0.0
0.00 1:000
10 (rigtt shifl, ettreme)
29.4-1.02
25.0+3.02
0.00 +0.00
0.00 +0.00
19.2
18.8 188-+1.00
0.00 +0.00
0.00 +0.00
À 0.5 (oompeession
0.9 00.00
102-+0005
0.52 ::000
6.01 10.00
0.20 -0.03
10.04
0.71 +:000
0.01 1.001
Scale
A 3.0 (dilation, maserate)
004-00.2
0.04 +0.1
0.21 +0.6
0000 +0:00
0.06 00.01
0.06 +0.01
0.34 +0.07
8.00
2 255 (diiotion, extreme)
0.0 =0.02
0.04 +0.01
+0.0
0 +0.00
0.06 00.01
0.05 +0.01
0.38 +0.07
+0.00
A dkf shift strong dilation)
2.18-0.12
2.21 +0.11
0.23 +0.0
8J00 00.00
1.24 00118
+0.17
0.38 +0.07
8,00 9-000
Affine
5. (shift stnnng compecssios)
748-3.04
67.9 +5.02
4.00 +0.15
0.24 +0.03
77.5 77.55
73.5 +6.92
4.97 +0.31
0,18 +0.01
5.A. (hin dilation modente)
=0.61
7.69 #0.28
0.21 =000
0.00 $00.00
4.42 =0.15
4.28 +0.02
0.34 =0.07
0.00 960000
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Published as a conference paper at ICLR 2026
Table 12: Detailed model performance (Accuracy in % mean + std over 5 seeds) for different normalization strategies. For each row (perturbation), the best result within each normalization group is highlighted in gray. The overall best result is in bold.
Dropout
Model
Corruption
Stand.
SEqSI
AftEq
Stand.
SE
SEqSI
AffEq
Original
91.7 #0.3
91.3 #0.1
91.2 +:03
89.6 =0.1
91.3 #0.2
91.6 #0.2
91.7 +0.3
81.2 =0.6
Affine transtormations (Aff.)
51.1 #100
53.4
91.2
89.6 =0.1
54.5 +0.8
58.6 +1.
+0.3
00.6
Scale (<
68.0 +0.8
91.3 +0.1
91.2 +0.3
89.6 +0.1
71.1 +1.0
91.6 +0.1
91.7 +0.3
81.2 0066
Scale (> 1)
91.0 #0.1
91.3 #0.1
91.2 +0.3
89.6 89.6 =0.1
87.4 #0.7
91.6 20.2
91.7 40.3
81.2 =0.0
Affine
64.1 =0.9
65.1 =0.7
91.2 20033
89.6 =0.1
63.5
=0.8
68.2 20.5
91.7
40.3
81.2 00.6
Non-Affine transformutions (NAfT)
Shift saturated
72.0 =0.6
73.4 104
79.0 100.6
76.7 =0.4
73.2
20.4
76.0 20.2
79.6 +0.3
71.4
Scale (> 1) saturated
78.1 +0.5
78.3 +0.66
77.6 +0.5
76.0 0.55
77.9 +0.4
79.0 +0.4
79.4 +0.2
72.1 0.55
Affine saturated
59.9 =0.8
59.6
72.3 #1.0
70.9 0055
61.2 +0.9
62.5 0066
74.1 20.6
=0.8
Sputially-vayying Affine
31.8 =10
+23
12.5
63.6=10
39.9 =1.3
44.4
73.8
42.6 20.8
Contrast Inversion
9.6 0.77
15.4 +2.0
56.5 +1.6
52.9 110
13.9 +0.4
41.7 43.7
58.7 4-0.9
37.1 11.2
Noise (low)
90.8 10.2
90.6 +0.2
90.1 10.6
88.5 -0.2
90.7
=0.1
90.8 20.2
90.6 +0.2
80.0 -0.7
Noise (high)
22.4
15.2
20.3
18.4
+0.4
+2.5
Gamma (darken)
74.3 =1.0
76.4 00.7
79.9 =0.4
76.4 00.5
77.1 =0.6
79.8
00.4
80.7 #002
70.5 00.9
Gunma (lighten)
85.9 #0.6
85.3 00.7
90.3 +002
88.6 =0.1
87.0
20.3
87.3
20.3
90.9 +0.2
80.4 20.6
BN
Model
Corruption
Stand.
SE
SFq81
AflFq
Stand.
SE
SEqS1
AflFi
Original
94.2 #0.1
94.0
+0.2
93.9 +0.3
91.6 0.22
95.7 #00.1
95.7
+0.2
95.7 +0.2
94.3 +0011
Affine transformations (Aff.)
Shift
91.6 =0.2
91.7 :0.2
93.9 10.3
91.6 91.620.2 =0.2
40.3 11.7
39.1 110
95.7 10.2
34.1 20.5
Scale (<)
94.1
93.9
93.5
90.8
79.2
Scale (>>]]
94.2 #0.1
94.0 =0.2
93.9 +003
91.6 0022
64.5 #3.7
62.0
#3.9
86.0 +1.
50.2 2266
Affine
90.9 10.1
91.2 =0.2
93.8 1:033
91.4 =0.2
34.2
12.1
33.4 118
84.4 +1.1
25.1 20.9
Non-Affine transformations (NAIT.)
Shift saturated
79.9
80.1
78.2 0.5
77.9
+0.7
+0.6
Scale saturated
79.7 :0.6
79.7 +.0.3
79.0 +0.2
76.5 0066
79.2 +0.9
79.6 +0.2
78.8 +0.5
76.8 20.8
Affine saturated
72.3 +0.8
72.8 +0.55
73.3 +0.4
71.1 0.44
67.1
+0.6
67.3
+0.5
74.3 +0.5
65.1 =0.6
Spatially-varyine Affine
86.5 #0.6
86.8 +0.3
93.4 1022
40.7 =1.1
47.5 11.1
47.4 112
71.9 +100
38.6 -0.9
Contrust Inversion
38.4
+0.9
67.6
61.6
10.8 =1.1
10.8 20.4
67.9
10.2
Noise (low)
92.4 10.1
92.2 +0.
91.5
0.22
92.8 10.3
92.8 10.1
92.2 10.4
91.2 -0.2
Noise (high)
28.6 11.7
27.8 11.7
19.3 +1.3
32.2 1.0
13.3
+1.
12.7 +0.7
12.8 40.9
13.3 225
Gamma (darken)
85.0 +00.5
85.2
84.7
82.5 +0.4
Gamma (lighten)
93.2 #0.1
93.1
20.3
93.2 10.3
90.7 00.1
92.7 #0.1
92.8 #0.3
94.6 4:0.1
90.6 =0.4
LN
Mix
Model
Corruption
Stand.
SEgS1
Aflli
Stand.
SE
SEgq1
Afiig
Original
93.3 #0.1
93.5 +0.3
94.1 ::0.1
92.4 =0.3
95.4 =0.1
95.4 10.1
95.5 E0.2
=0.1
Affine transtormations (Aff.)
Shift
56.4 +0088
59.5 +0066
94.1 +0.1
92.4 +0.3
94.6 +0.1
94.6 +0.11
95.5 402
93.7 #0.1
Scale (<) 1)
79.2 +0.9
92.9 10.3
91.8 10.4
92.2 2033
95.2 #0.1
95.2
20.
94.6 +0.2
93.6 =0.2
Scale (>
92.4
93.5
+0.3
95.4 +0.1
95.4
Affine
66.5 #1.1
+0.8 68.3
93.5 +:0.2
92.4
=0.3
93.2
#0.2
#0.1 93.4
95.2 +:0.1
93.7 =0.2
Non-Affine transformations
(NAff.)
Shift saturated
73.8 11000
77.7
+0077
80.9
1:055
79.2
0.7
81.4
10.4
81.7 10.3
81.6
4-.33
79.2 20.4
Scale (> saturated
78.9 =0.4
79.5
+0.7
80.1 1:0.3
77.8
0066
79.5
10.6
79.8 +0.4
79.8
40.4
76.3 20.8
Aftine saturated
66.3 +1.5
68.0 +0.8
74.3 +0.6
72.2
1.1
73.4
+0.8
74.2 0.9
74.2 +0.4
70.5 +066
Spatially-vary Affine
47.3 +2.0
51.8 +2.5
82.6 1:0.9
73.5
-32
89.4
+0.3
89.5 20.2
94.9 +0.2
44.6
Contrust Inversion
11.4
10.4 12.4
63.4 10.88
55.1 22.6
66.7 10.3
67.2 20.5
71.7 40.4
66.8 -0.8
Noise (low)
91.1 +0.3
91.7 +0.3
92.1 +0.2
91.2 0.3
93.1
=10.1
93.1 :0.1
92.3 10.3
-0.1
Noise (high)
15.5 =0.6
18.0
13.941.0
23.1 -3.0
13.6
=1.5
13.7 10.7
12.2 40.4
21.0
Gamma (darken)
75.7 +0.4
80.4 +0.7
83.7 +0.4
81.8 +0.6
86.6 +0.1
86.4 +0.2
86.3 +0.4
82.6 +0.4
Gamma (lighten)
90.9 +0.3
91.2
93.3
91.7 =0.3
94.8 =0022
94.8 20.1
94.8 10.1
92.9 00.2
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C.8 DEMONSTRATING SCALABILITY ON MORE COMPLEX TASK: OXFORD- IIIT PETS AND STANFORD CARS CLASSIHICATION
To demonstrate the scalability of our SEySI architecture to more complex and resoetoreoolution computer vision tasks, we conducted experiments on the Oxford-IIIT Pet (Parkhi et al. 2012) and Stanford Cars Krause et al. 2013) datasets.
Scaling the Challenge: From CIFAR-10 to Oxford-IIIT Pets and Stanford Cars. The transi- tion from CIFAR-10 (Krizhevsky al. 2009) to the Oxford-IIIT Pets and Stanford Cars datasets represents a significant increase in difficulty, providing robust test for scalability:
Image Resolution:
The input images are scaled from 32 x 32 pixels for CIFAR-1O to 224 x
224 pixels, which is nearly 50-fold increase in the number of pixels. Image resolution can vary, but all images are resized to 224 x 224 pixels. This requires an architecture capable of handling much larger receptive field and spatial hierarchy.
Task Complexity: The classification task becomes significantly more fine-grained moving from 10 general object classes CIFAR-10 37 distinct pet breeds Oxford-IIIT Pets, and an even more challenging 196 classes of car models (differentiated by make, model, and year) in Stanford Cars.
Image Variability: Both datasets feature greater intra-class variation in object scale, pose, and background clutter compared CIFAR-10, posing a greater challenge for generaliza- tion.
Scaling the Architecture: From ResNet-20 ResNet-18. To accommodate this increase in scale, we adapted the backbone of the model from a ResNet-20-like structure to a standard ResNet- 18 architecture (He et al. (2015), a common choice for processing 224 224 images. The core principles of our model families (Standard, SEq, SEqSI, AffEq) are maintained, but the underlying architecture scaled up as follows:
Aggressive Initial Downsampling: The initial layer is adapted for high-resolution inputs. It uses a large 7 convolution with a stride of 2, followed by a 3 max-poollll max--oooling layer with a stride 2. This combination efficiently reduces the spatial resolution from 224 x 224 56 56, standard practice to manage computational cost and build hierarchy of features.
Deeper and Wider Structure: The network features four residual stages instead of three. The number of channels progresses from 64 to 128, 256, and finally 512, creating a deeper and wider feature hierarchy suitable for more complex task. The spatial resolution is successively halved at each stage, down to a final 7 x 7 feature map.
The classification head operates the 7 x 7 x 512 feature map. After global average pooling, the resulting 51 512-mmmensoon vector is projected to the appropriate number of classes (37 for Pets, 196 for Cars).
Adapted Classification Head:
Crucially, the fundamental design of each model family remains unchanged. For instance, the SEqSI model still employs a shift-invariant convolution its first layer and fias-free convolutions through- out the rest of the network. While the overall structure is scaled up, the internal design of the residual blocks and the application of architectural constraints are conceptually identical those used for the CIFAR-10 experiments (see Appendix C.I).
Results and Discussion. The primary objective of these experiments on the Oxford-IIIT Pets and Stanford Cars datasets was not achieve state-of-theeart accuracy, but rather to demonstrate the scalability of our constrained architectures. For these comparisons, it is important to consider the Standard model as the bascline, as represents an unconstrained CNN. The key point is that both SE and AffEq models, despite their significant architectural constraints, preserve their ability to learn the complex and fine- -in-ggraied features required for these challenging high-resolution tasks.
The results, presented in Table 13 and Table 14 serve as strong evidence for this. First, they confirm that the theoretical guarantees of our models are preserved when scaled to a deeper ResNet-18 backbone: both SEgSi and AffEi maintain perfect invariance affine transformations, feat the
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Standard models cannot achieve even with data augmentation. More importantly, this robustness does not come at the cost of learning capability.
On the Oxford-IIIT Pets dataset (Table 13p, without data augmentation, SEqSI (44.4% accuracy) is highly competitive with the Standard model (44.5% accuracy) on original images, while demon- strating superior robustness to affine corruptions (e.g.. 44.4% for SEqSI 12.4% for Standard under a shift). With data augmentation (Train Aug. = All), SEqSI (31.7% accuracy on original images) largely outperforms all other models, including Standard (14.6% accuracy), especially on non-affine transformations like "Spatially-varying Affine" (25.2% for SEqSI vs 10.0% for Stan- dard).
For the even more complex Stanford Cars dataset (Table 14). both SEqSI (33,2% accuracy) and AffEq (44.8% accuracy) significantly outperform the Standard model (27.3% accuracy) on clean data without any augmentation. With data augmentation (Train Aug. All), SEqSI (22.8% % accu- racy) and AffEq (20.8% accuracy) remain clearly superior to Standard (6.9% accuracy) on orig- inal images, and maintain this advantage across various corruptions (e.g.. for "Gamma (darken)", SE achieves 13.5% and AfʃEq 12.8% compared to Standard 5.1 accuracy).
It is important to note two key observations from these experiments:
The significant drop in performance for all models when trained with strong data augmen- tation (Train Aug. All, using parameters from Appendix C.2) suggests that the photo- metric perturbations applied for augmentation are too aggressive for these higher-resolution datasets, This motivated our complementary study using more moderate perturbations, de- tailed in Appendix C.9
In the most complex scenario of the Stanford Cars dataset, AffEq consistently shows a clear advantage, often achieving the highest accuracy despite being the most constrained network. This suggests that the architecture properties can be beneficial for generalization and convergence on challenging tasks.
These findings validate the scalability of our approach. In conclusion, these findings validate that the principles behind SEqSI and AffEq are indeed scalable. They can be successfully integrated into standard, deeper backbones tackle complex, real-world computer vision problems, offering a robust and principled alternative to standard architectures. The absolute accuracy scores should be viewed in the context of a relatively shallow ResNet-18 model trained for a limited number of epochs (200). The crucial point is the demonstrated ability of these constrained models to learn effectively at scale while providing certified robustness. This is further supported by experiments with more moderate perturbations, as detailed in Appendix C.9
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Table 13: Robustness to photometric corruptions the Oxford-IIIT Pet test set. Test accuracy (%) (mean + std over 5 seeds) of four ResNet- architectures. Models are grouped by the training augmentation strategy used: none (Ø), affine (Aff.), non-affine (NAff.), all combined (All). Each row corresponds to a different perturbation applied at evaluation. Within each training strategy (column group), the best result is highlighted in gray. The overall best accuracy for each perturbation is in bold. Models were trained for 200 epochs.
Train Aug. !
Train Aug, Aff
Model
Corruption
SEq
SEgS1
Affli
SEq
S1g811
Amig
Original
44.5 #1.5
35.9 #0.7
44.4 +1.3
=0.9
24.8
16.7
26.6
#4.8
35.9 #2.1
32.5 #0.4
Affine transformations (Aff.)
Shift
12.4 +00.7
10.3
+0.3
44.4
F1.3
38.1 0099
23.5 +6.6
25.3 +4.88
35.9
32,5 +0.44
Scale (< 1)
28.1 +0.7
35,9 +0.7
44.4
+1.3
38.2 0099
22.7 16.3
26.6 14.
35.9 42.1
32,5 20.4
Scale (> 1)
37.9 +0.4
35.9 +0.7
44.4
+1.3
38.1 0.99
24.5 +6.7
26.6
+4.88
35.9 +2.1
32.5 +0.4
Affine
15.9 #0.9
15.8
20.4
44.4
+1.3
38.2
00.9
22.3
-6.3
25.1
14.3
35.9 4:2.1
32.5 =0.4
Non-Affine transformations (NAIL.)
Shift saturated
22.0 =0.4
18.5 +0.7
28.9 40.8
27.0 20.8
17,9
+4.9
19.0 #3.1
23.7 11
23.5 +0.8
Scale (> 1) saturated
27.6 110
22.7 +0.4
27.9 +0.9
27.7 00.4
18.1
+4.5
18.7 +3.3
23.6 +0.7
23.7 0.2
Affine saturated
16.2
17.5
Spatially-varye Affine
6,2 =0.5
7.4 #0.6
31.1 #1.8
16.4 00.9
16.8 =4.7
22.4 24.0
25.3
13.8 110
Contrust Inversion
2.4 10.3
2.6 10.3
6.9 10.4
5.8 =0.3
10.6
3.7 20.2
6.3 10.66
4.8 -0.6
Noise (low)
35.9 +0.7
+0.9
24.6
44.8
Noise (high)
29.6:11
28.0
5.3
16.1 23.4
13.5
+3.9
16.1 +2.9
3122
16.2 114
Gamma (darken)
18.8 +0.5
16.5 +0.5
21.5 +1.0
21.5 0.5
13.8
+3.6
14.1
+2.0
17.5 +1.3
19.0 0.5
Gamma (lighten)
31.6 =12
24.1
+0.8
38.0 +1.0
35.1 11.
20.2
45.5
22.6
+3.9
30.2 11.8
29.8 =0.5
Train Aug.
NAIf
Train Aug. All
Model
Corruption
Stand.
SEg
SE4SI
Afllg
Stand.
SEq
SE&SI
Afiig
Original
20.0 +8.11
18.4 +3.7
34.7 +1.1
30.1 =1.3
14.6 +33.4
13.9 +2.0
31.7 +1.88
25.6 #:2.3
Affine transformations (Aff.)
Shift
13.2 =3.9
13.7 13.7 118
34.7:111 E1.1
30.1 113
13.4 +3.1
12.0 11.0
31.7 E18
25.6 =2.3
Scale <<
14.9 16.0
18.4 13.
34.7 +1.1
30.1
1.3
12.2 12.6
13.9 12.0
31.7 11.8
25.6 -2.3
Scale (>
17.6 17.8
18.4 13.7
11.1
30.1 1.3
14.2 -3.5
13.9 12.0
31.7 +1.8
25.6
Affine
12.8 14.55
14.8 #23
34.7 +1.1
30.1
1.3
#3.1 12.7
12.3 +1.7
31.7 +:1.8
25.6 42.33
Non-Affine transformations
(NAm.)
saturated
16.8 =6.5
14.9 +2.8
20.88
24.5
00.6
11.9
+2.5
11.2 +13
24.4 11.5
20.6
Scale saturated
16.7 16.6
16.1 +2.9
30.3 11.0
26.3 1.4
12.6 -2.7
11.6 11.
26.0 +1.2
22.0
Afline saturated
14.9 +6.3
14.3 422
28.2 +1.2
25.1 1.6
11.1 +2.0
10.7 +1.4
24.2 +0.9
20.9 1.7
Sputially-varying Affine
10.7 +3.1
11.3 115
29.7 +1.0
18.6111
18.6
10.0 +2.1
10.8 +1.6
25.2 41.3
11.1
Contrast
5.8
Noise (low)
20.0 =8.2
18.3 +3.7
+1.0
30.1 1.3
#3.4
13.9
31.5 42.0
25.6 =2.3
Noise (high)
18.0 +8.2
16.0 1300
30.2 11.7
17
11.8 42.6
10.9
25.1 42.3
23.2 119
Gamma (darken)
15.9 16.2
12.5 +2.3
21.6
20.6
10.8
9.1
18.8 40.9
11.4
Gumma (lighten)
18.1 18.2
17.2 232
29.3 11.2
13.2
12.8 12.0
28.7 +1.7
24.7 118
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Published as a conference paper at ICLR 2026
Table 14: Robustness to photometric corruptions on the Stanford Cars test set. Test accuracy (%) (mean + std over 5 seeds) of four ResNet-18 architectures. Models grouped by the training augmentation strategy used: none (O), affine (Aff.), non-affine (NAff.), or all combined (All). Each row corresponds to different perturbation applied at evaluation. Within each training strategy (column group), the best result is highlighted gray. The overall best accuracy for each perturbation in bold. Models were trained for 200 epochs.
Train Aug.
Aug.
Model
Corruption
Stand.
SEq
SEqSI
AffEq
Stand.
SEq
SEqSI
AffEq
Original
27.3 #3.4
24.9 #2.0
33.2 +2.7
44.8 1.44
18.0
#5.2
18.5 #1.4
24.7 #2.8
25,3 =0.0
Affine transformations (Aff.)
Shift
4.6 =0.7
4.5 +0.2
33.2 12.7
44.8 1.4
17.6 15.2
18.1 +1.5
24.7 42.8
25.3 00.6
Scale (<)
24.9
33.2 +:2.7
44.8
16.4 +4.8
18.5 +1.4
+2.8
Scale (>
24.6 #3.3
24.9 #2.0
33.2 +27
44.8 1.44
17.6 #5.1
18.5 114
24.7 +2.8
25.3 =0.6
Affine
7.4 1.00
7.5 20.5
33.2 +2.7
44.8
1.4
16.9
=5.0
17.7
11.5
24.7
42.8
25.3 -0.6
Non-Affine transformations (NAIT.)
Shift saturated
10.6 :16
10.1 #0.9
21.7
12.1 +3.5
12.5 #10
16.3 +1.
Scale saturated
17.7 +2.3
16.1
21.4 +1.8
31.9 0.9
12.2 +3.1
12.4 +0.9
16.6 +2.3
18.4 +0.6
Affine saturated
3.8 10.5
3,9
20.0
11.8
29.4
00.8
11.2
+3.0
11.2 40.9
15.2 42.0
17.1 =0.5
Spatially-varing Affine
=0.5
2.2
28.8 424
36.3 110
14.0
14.3
16.7 11.
21.0 11.9
18.7 =0.5
Contrast Inversion
0.5 +0000
0.5 +0.0
1.31
1.5
+0.1
0.8 +0.1
1.1 +:0.1
3.4 003
Noise (low)
27.3 -3.4
21.9
33.0
44.6 1.3
17,9
15.2
18.5 +1.
24.6 42.
25.2 00.6
Noise (high)
11.8 +1.6
+1.
7.0
21.4
+24
10.6
+1.7
11.3 +0.7
7.0 +2022
15.2 1.0
Gamma (darken)
11.1 =1.
903
17.
25.7 20.7
10.1
12.6
20.4
12.8
14.4
Gumma (lighten)
17.1 225
11.
22.5
42.2
111
16.1
14.6
16.7
11
21.8 42.7
23.7 00.5
Train Aug. NAI
Train Aug. All
Model
Corruption
Stand.
SEq
SEqSI
AftEq
Stand.
SEq
SEqSI
AftEq
Original
6.1 +4.6
17.4 +1.9
32.0=1135
31.0 #14
6.9 #4.4
12.6
#2.9
22.8 #3.1
20.8 1.9
Affine transformations
(Aff.)
Shift
4.8 +3.55
13.2 41.7
32.0 1.3
31.0 11.
6,4 14.
11.8 12.7
22.8 13.1
20.8 +1.
Scale
5.0
17.4
32.0+1.3
31.0 1.4
+3.9
12.6
22.8
11.9
Scale (>
5.7 +4.3
17.4
+1.5
31.0 +1.4
6.6 +4.3
12.6 +2299
22.8 +3.11
20.8 +1.9
Affine
4.7 13.4
13.6
32.0 11.5
31.0
114
6.0
13.
11.7 12.5
22.8 4:3.1
20.8
1:1.9
Non-Affine transformations
(NAf.)
Shift saturated
5.0 +3.6
13.4
24.8 11.3
23.9
110
5.4 L33
9.5 12.2
17.2 12.2
16.2 114
Scale saturated
5.0 13.7
14.4
27.3 114
26.1
114
13.4
12.4
11.3
Affine saturated
4.6 +3.33
13.0++1.5
25.7 +1.4
24.4
113
5.2 +33
+1.9
17.5 +-2.3
16.3 +1.3
Spatially-varye Afline
4.2 1:3.1
11.9
29.8 -1.4
28.2 111
5.4 135
10.8 12.4
19.8 12.7
17.4 1118
Contrast Inversion
4.8.
13.7
11.9
22.9 11
122
3.8 11.5
3.8 10.9
12.0 1.2
Noise (low)
6.0
32.0
31.0
6.9
Noise (high)
5.5
15.9
11.
25.2 111
6.3
10.8 12.3
19.4
16.6 11.5
Gamma (darken)
4.743.4
10.9
21.3
19.1 110
5.1
7.5 41.5
13.5 11.7
12.8 +13
Gamma (lighten)
5.8 44.4
17.1 41.9
31.4 1..4
30.9
113
6.5 142
11.9 42.5
21.7 43.0
20.3 +18
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Published as a conference paper at ICLR 2026
C.9 DETAILS ON 'MODERATE' PHOTOMETRIC PERTURBATIONS FOR SCALABILITY EXPERIMENTS
The photometric transformations used for the scalability experiments Oxford-IIIT Pet and Stan- ford Cars are detailed below. The parameters were adjusted to a "moderate" intensity, less ag- gressive than those for CIFAR-10 (Appendix C.2 better assess model performance on these higher-resolution datasets.
Transformation Groups. Transformations are categorized into two main groups:
Affine Transformations (Aff): This group consists of 4 non-saturated global affine intensity trans- formations.
Global Additive Shift: T(x) x + μa, with μr 0[
Global Scaling (Compression): T(x) Aæ, with 4(00
Global Scaling (Dilation): T(z) Aa, with À ~ (1.00, 2.0j.
Global Affine: T(x) = Aæ + j4, with X ~ [0.5 2.0] and μ ~ ~~ 211-0.,
Non-Affine and Saturated Transformations (NAff): This group includes transformations that cither non-affine, spatially-varying saturated.
Saturated Additive Shift: T(x) = clip(æ + 0, 1), with ~ -
Saturated Scaling: T(x) = clip(Aæ 0, 1), with 4[1.0,
Saturated Affine: T(x) = cAip + u, 1), with À ~ 4(0.8, and μ 4[-002
Spatially-Varyine Affine (Linear): (u,)
((u,v)a(u, p(u,v), with
A(u, ~ 2405. 1.0] and ((u, v) ~ 4[-0.5, 0.5] for each pixel (u, v).
Contrast Inversion (Linear): A spatially-varying scaling (ux) v) = M(u, v),(u, v), with A(u, v) ~ ([1000 -0.5 for each pixel (u,
Additive Gaussian Noise (Low): T(x) clip(" + 1), where
o² and
σ ~ -0000 0.033.
Additive Gaussian Noise (High): T(x) clip(x + n, 0, 1), where .N(0, and
~ 100,
Gamma Correction (Lighten): T(x)) = x², with (000.,,1.0) Gamma Correction (Darken): T(x) = x², with y ~ 41110,, 2.5).
Validation of the results. The results of these experiments, presented in Tables 15 and 16 show that the 'moderate' data augmentation strategy is better suited for these datasets, leading to improved overall performance. For instance, the performance of the SEgSI model trained with all augmen- tations (Train Aug. = All) on the original test data improves significantly. increasing from 31.7% (Table [3) to 45.7% on Oxford-IIIT Pets, and from 22.8% (Table [4] 37.0% on Stanford Cars. Crucially, the observations regarding the superior performance and robustness of SEqSI and AffEq models remain consistent, confirming the conclusions from Appendix C.8 about the scalability and validity of our approach.
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Published as a conference paper at ICLR 2026
Table 15: Robustness to 'moderate' photometric corruptions the Oxford-IIIT Pet test set. Test accuracy (%) (mean = std over 5 seeds) four ResNet-18 architectures. Models are grouped by the training augmentation strategy used: none (0), affine (Aff.), non-affine (NAff.), all combined (All). Each row corresponds to different perturbation applied at evaluation. Within each training strategy (column group), the best result is highlighted in gray. The overall best accuracy for each perturbation is in bold. Models were trained for 200 epochs.
Train Aug. !
Trai Aug, Aff
Model
Corruption
SEq
SEgS1
Affli
SEq
S1g811
Amig
Original
44.5 #1.5
35,9 +0.7
44.4 +1.3
38.2 #0.9
41.4 #3.9
40.6
+1.
43.0 E1.4
37.6 =0.5
Affine transformations (Aff.)
Shift
43.4 +1.5
35.1
+0.8
44.4 F1.3
38.2 0.99
41.4 +3.8
40.5 +1.6
43.0 F1.4
37.6 +0055
Scale (< 1)
44,4 114
35,9 -0.7
44.4
+1.3
38.2 0.99
41.4
+3.9
40.6
116
43.0 +1.4
37.6 -0.5
Scale (> 1)
44.2 +1.4
35.9
+0.7
44.4
+1.3
38.2 0.9
41.3
+4.0
40.6
+1.7
43.0 +1.4
37.6 +0.5
Affine
43.3 #14
35.1
+0.8
44.4
11.3
38.2
00.9
41.3
14.0
40.4
#16
43.0
37.6 20.5
Non-Affine transformations (NAIL)
Shift saturated
43.4 +1.5
34.8 +0.7
44.2 +1.3
38.0
00.9
41.4
+3.9
40.2 11.7
42.7 +1.1
37.4 20.6
Scale (> 1) saturated
44.1 11.
35.5 +0.88
44.0
11.4
37.9
0.8
41.1
+3.8
40.3
11.7
42.6 +1.2
37.6 +0.6
Affine saturated
25.7
Spatially-varye Affine
11.
35.3 #0.6
44.5 #1.4
38.1 =0.8
41.3
#4.0
40.6 #1.6
42.9
37.6 =0.5
Contrust Inversion
2.7 20.5
2.6 203
10.5
5.8 =0.2
4.5 20.4
10.6
7.1 10.6
5.5 =0.4
(low)
35.9 +0.7
+0.9
+1.5
Noise (high)
42.2 112
34,9 +10
244
32.5 =23
37.1 25.5
11.5
21.4 42.1
33.0
Gamma (darken)
41.0 +1.6
33.8 +04
41.9
36.5 +0.9
39.5
+4.1
38.9 +2.1
40.8
+1.66
36.0 +0.6
Gamma (lighten)
43.5 =1.
34.4
#10
44.0 +1.0
38.2 00.9
41.0
13.8
39.9 116
42.6 116
37.4 20.6
Train Aug.
NAIf
Train Aug. All
Model
Corruption
Stand.
SEg
SE4SI
Aflly
Stand.
SEq
SE&S1
Afiig
Original
42.4 116
31.7 +:4.5
44.2 +1.5
33.8 11.2
37.6
+:4.5
34.1 +2.3
45.7 +-0.9
35.2 #066
Affine transformations (Aff.)
Shift
42.2 +1.
31.7 31.7 14.
44.2 11.5
33.8 12
37.6 14.5
+2.4
45.7 +0.9
35.2 20.6
Scale (<
42.4 =18
31.7 14.
44.2
33.8 12
37.7 14.5
34.1 12.3
45.7 +0.9
35.2 20.5
Scale (>
42.2 =1.
31.
41.2
33.8
37.6 215
34.1 12.3
45.7 10.9
35.2 20.6
Affine
42.3 #1.6
14.7
44.2 +1.5
1.2 33.8
+4.5 37.5
+2.2 34.0
45.7 +009
35.2 0.6
Non-Affine transformations (NAfT)
saturated
42.4 118
31.6
44.0
33.7 1.2
37.7
14.4
34.1
+2.2
45.5 +0.9
=0.4
Scale saturated
42.3 116
146
41.
33.8 1.2
37.4
14.
34.0 12.3
45.5 10.9
35.2 20.7
Affine saturated
32.2 1.5
23.4 +3.5
+1.2
27.7 009
26.9
+3.8
22.6
+1.9
35.0 +1.0
28.5 0.6
Sputially-varying Affine
42.5 116
31.7 14.7
44.2 11.5
33.6
37.8
14.5
34.1 424
45.7 40.9
35.2 20.6
Contrast
29.4
Noise (low)
41.4
31.7 246
44.1 #1.6
33.7 1111
37.7 24.5
34.1 22.3
45.7 10.9
35,3 =0.6
Noise (high)
42.0
30.8 148
41.7
11.
34.
32.8
11.
44.2 +1.5
34,5 205
Gamma (darken)
41.8
30.6
42.8
110
34.6
32.5
121
44.2 11.0
34.1 20.7
Gumma (lighten)
41.3
31.8
41.9
33.7 1.0
37.6
12.2 33,9
45.4 40.8
35.3 203
36

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Published as a conference paper at ICLR 2026
Table 16: Robustness to 'moderate' photometric corruptions the Stanford Cars test set. Test accuracy (%) (mean = std over seeds) of four ResNet-18 architectures. Models are grouped by the training augmentation strategy used: none (0), affine (Aff.), non-affine (NAff.), all combined (All). Each row corresponds to a different perturbation applied at evaluation. Within each training strategy (column group), the best result is highlighted in gray. The overall best accuracy for each perturbation is in bold. Models were trained for 200 epochs.
Train Aug. !
Train Aug, Aff
Model
Corruption
SEq
SEgS1
Affli
SEq
S1g811
Am3g
Original
27.3 #3.4
23.3 +2.7
33.1 +2.7
44.8 =1.4
19.9 22.4
26.7
20.7
33.1
42.1
41.1 =1.1
Affine transformations (Aff.)
Shift
14.0 +:2.3
12.4 +1.2
33.1 +2.
44.8 11.4
19.7 +2.
26.3 +0.7
33.1 42.1
41.1
Scale (< 1)
25.7 #3.1
23.3 +2.7
33.1 +2.7
44.8 11.4
19.5 +2.3
26.7 10.7
33.1 42.1
41.1
Scale (> 1)
26.4 +3.4
23.3 +2.7
33.1
+2.7
44.8
19.8
+2.2
26.7
+0.7
33.1
+2.1
41.1
Affine
15.6 22.4
14.1 113
33.1
4:2.7
44.8
1.4
19.4
+2.22
26.5
20.6
33.1
12.1
41.1 1111
Non-Affine transformations (NAIT.)
Shift saturated
19.9 +2.9
17.4 +1.
30.4 +224
41.9 1.
18.5 #2.1
25.0 +0.6
30.5 +2.1
38.5 2009
Scale (> 1) saturated
24.1 +3.3
20.5 +2.5
29.0
40.6 1111
17.6
24.1 +0.8
29.3 +20
37.0
Affine saturated
20.1
Spatially-varye Affine
13.7 +2.2
11.2
32.9
44.3 1.2
19.4 +2.3
26.3 #0.8
32.8
40.5 1.0
Contrust Inversion
0.5 10.1
0.5 10.0
1.2
7.5 114
0.9 10.1
1.2 10.2
122022
6.3 20.7
(low)
27.3
23.4 +27
32.9
19.9
+0.7
32.8
Noise (high)
19.7 =24
16,9 +22
16.6
33.4
13.4
19.2 20.7
16.5 43.6
11.
Gamma (darken)
19.1 +26
15.1 +2.0
26.6 +2.3
37.5 1.2
16.3
+1.9
21.3 +0.7
26.6
34.3 +066
Gamma (lighten)
23.6 +3.3
20.2
118
32.2 42.6
44.2 12
19.0
:2.1
26.2
20.7
32.2 42.4
112
Train Aug. NAIf
Train Aug. All
Model
Corruption
Stand.
SEg
SE4SI
Aflig
Stand.
SEq
SU&SI
Afiig
Original
13.5 #1.4
22.3 +1.8
35.0 +2.3
40.3 11.8
14.6 +1.00
23,9
+2.7
37.0 -1.5
40.2 -0.4
Affine transformations
(Aff.)
Shift
12.7 +13
21.1 #16
+2.3
40.3 118
14.2 +0.9
23.2 +2.5
37.0 +1.5
40.2 20.4
Scale <<
12.7 11.
22.3 :18
35.0
40.3 1.8
14.2 10.8
23,9 12.7
31.0
20.4
Scale
13.0 =13
22.3 118
35.0 12.3
40.3 1.8
14.3 =1.0
23,9 12.7
31.
40.2 =0.4
Affine
12.2 +1.2
21.6
35.0
40.3 1.8
20.9 14,0
23.1
#2.5
37.0
40.2
00.4
Non-Affine transformations
(NAm.)
saturated
12.8 =12
20,9 #1.
43.44
38.1 2.0
13.8
+0.9
22.6
+2.6
35.1 +1.3
37.9 20.4
Scale saturated
12.1 =12
21.
33.2
1..
13.1 20.9
21.9
1266
35.1
37.8
-0.3
Afline saturated
9.3 1.1
+1.6
26.8
31.0 -2
9.9 +0.8
17.1 +22
21.2
30.1 =0.4
Sputially-varying Affine
12.3 113
21.0
42.4
39.8 -2.0
13.9 11.0
22.9
42.6
41.4
40.0 -0.6
Contrast Inversion
Noise (low)
11.4
+18
42.0
40.0 11.
14.5 =1.
23,9 +2.7
31.5
39,9 =0.4
Noise (high)
12.4
20.5 117
32.8 42.4
35,5 15
13.2 10.8
21.8 42.2
34.0 +1.2
35.1 -0.6
Gamma (darken)
11.1
115
29.4
33.7
11.1
19.6
31.1
33.6 -0.3
Gumma (lighten)
11.1
22.1
348424
40.3
14.3 111
23.6 12.7
36.8 +1.5
40.2 203
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C.10 TRANSFERT LEARNING FROM STANDARD MODEL
Experimental Setup. The transfer learning experiment was conducted on the Stanford Cars dataset (Krause al., 2013). We used ResNet-18 architecture with weights pre-trained on ImageNet- IK (ResNet @_Weigght IMAAEEETIK_V from PyTorch). The goal was to assess the practicality and compatibility of SEqSI architectures when fine-tuned from a standard model.
Model Adaptation from Pre-trained Weights. Since the pre-trained model is Standard archi- tecture, adapting it for our constrained models required significant modifications, creating a struc- tural handicap for SEq, SEqSI, and AffEq.
The pre- pre-trained weights were loaded directly. Only the final fully- colly-connected layer was replaced with a new one, randomly initialized, to match the 196 classes of the Stanford Cars dataset.
Standard:
SEq: All bias parameters were removed from the pre-trained layers to enforce scale-equivarianee. The final classifier was also replaced.
SEqSI: In addition to removing all biases like in SEq, the weights of the first convolutional layer were re-projected to be zero-sum. This was done by subtracting the mean from the pre-trained weight tensor of that layer, thus enforcing shift-invariance at the input.
AgffEq:
This model required the most drastic changes. All biases were removed, all ReLU activa- tion functions were replaced with 'SortPool', and the weights of every convolutional layer were re-projected to sum to one. These extensive modifications, which create a significant departure from the original learned representations, are the likely cause of the observed training divergence.
These adaptations highlight a key practical challenge: leveraging standard pre-trained models with architectures that have specific structural constraints.
Fine-Tuning Protocol. All models were fine-tuned for 200 epochs on the Stanford Cars training set using a batch size of 32. The learning rate was set to lc-2 for the Standard, SEq, and SEqSI models, and decayed using a cosine annealing schedule. For the AffEq model, a lower learning rate of 1e-3 was necessary prevent training divergence. For all models, the entire network was fine- tuned (no layers were frozen). The different convergence speeds noted in the main text (e.g.. 10 epochs for Standard vs. 50 for SEq/SEqSI to pass 60% accuracy) were observed during this process, with AffEq failing to converge to a meaningful accuracy level.
Results. The fine-tuning results, presented in Table 17, demonstrate the practical viability of SE- 9SI Despite the initial handicap from modifying the pre-trained weights. SE&SI proves its compati- bility with transfer learning by achieving competitive performance. reaches an accuracy of 82.3% on clean data, close to the advantaged Standard model (85.6%) Crucially, it provides certified ro- bustness to all affine corruptions where the performance of Standard model collapses (e.g., to 60.6% under simple shift), while also showing superior generalization on several challenging non-affine corruptions like "Spatially-Varying Affine" (82.1% vs. 67.2%) In contrast, the AffEq architecture fails to converge effectively in this transfer learning scenario, likely due to the extensive modifica- tions required to adapt the pre-trained weights, resulting in poor performance across all evaluation conditions.
Conclusion. This study demonstrates that SEgSI is practical and viable architecture for transfer learning. It can successfully adapt features from standard, pre-trained models. De- spite a structural modification leading slower initial convergence, it achieves competitive accuracy while preserving its architectural properties. In contrast, the AffEq architecture proves too rigid for this common and important workflow. This positions SEqSI as a principled yet pragmatic choice for fine-tuning pre-traines trained models in applications where photometric robustness critical.
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Table 17: Robustness in a transfer learning scenario on Stanford Cars. Test accuracy (%) of four ResNet-18 architectures fine-tuned from an ImageNet pre-trained model. Results are the mean + std over seeds. The best result in each group is in gray; the overall best is in bold.
Train Aug. = 0
Model
Eval. Perturb.
Stand.
SEq
SEqSI
AffEq
Original
85.6 0.2
82.1 #0.4
82.3 #0.1
37.3 11.5
Affine transformations (Aff.)
Shift
60.6 0.9
78.6 1.1
82.3 +0.1
37.3 1.5
Scale (< 1)
84.1 0.3
82.1 #0.4
$0.3
37.3 1.5
Scale (> 1)
66.2 +23
82.1
82.3
37.3 +15
Affine
49.4 +0.9
78.8 +1.0
82.3 +0.1
37.3 1.5
Non-Affine transformations (NAff.)
Shift saturated
81.1
79.2
80.2 10.4
35.0 114
Scale (> 1) saturated
81.1
79.2
79.3
33.8
Affine saturated
50.1
51.0
66.5
24.8
Spatially-varying Affine
1.3
7..9
82.1
36.2
Contrast Inversion
0.6
#.9
31.7
2.4
Noise (low)
80.3
81.9
81.9
36.1 #1.5
Noise (high)
13.2 +2.2
66.6
46.8 #2.6
3..6
Gamma (darken)
81.1 =0.4
76.3 #0.7
77.3 #0.5
31.4
Gamma (lighten)
84.5 +0.3
81.2 #0.4
81.9 #:0.2
36.6
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D EXPERIMENTS IN MACROMOLECULE CLASSIFICATION APPENDIX
D.1 DETAILS ON MODELS IMPLEMENTATION AND DATA PROCESSING
The original CZI challenge involves both localizing and identifying macromolecules. We simplity this to pure classification task on pre-extracted patches. This simplification is used below since: the full localization-ane ietiraattoonand-iide andiidentiiication pipeline is highly complex and often requires a separate, non- trivial post-processing algorithm for connected component analysis. By decoupling the tasks, we separate the identification module, allowing for a more direct and controlled evaluation our mod- els robustness. This is particularly important for assessing the performance of architectures like SE on the highly corrupted and variable cryo-ET images, enabling to draw clean conclusions about the benefits of our architectural properties under such challenging conditions.
Dataset Details. We use the dataset from the CZI cryo-ET challenge (Harrington et al. 2024). which consists of 7 tomograms containing 1, 1,269 annotated particle centroids belonging ent macromolecule classes. From these annotations, we extract 3D patches of size 64 x 64 x 64 centered on each particle. The dataset presents two main sources of variation. First, the challenge provides four different preprocessing pipelines for each tomogram, resulting in significant domain shifts for the same underlying data, as illustrated in Supp. Fig. 6 Second, there is substantial variability between the tomograms themselves due to different acquisition conditions (e.g. samples thickness, cryo quality...). Supp. Fig. 7 we show an example of this inter-tomogram variation. In Supp. Table [8] we summarize the voxel intensity distributions for each class across the splits, quantitatively confirming this significant variability.
differ-
These variations stem from three main sources:
Inter- preproceppoo variability: The dataset provides four different preprocessing pipelines for each tomogram, creating severe domain shifts as shown in Supp. Fig. 6 All tomograms are reconstructed from tilt series using WBP (Weighted Back Projection) as the base method (Peck et al., 2024):
WBPP reconstructed by Weighted Back Projection (no additional processing)
CTF deconvolved: WBP + local Contrast Transfer Function (CTF) correction (deconvo-
lution)
Denoised: WBP + DenoisET denoising
IsoNet Corrected: WBP IsoNet correction (denoising / messing-wedde compensation /
contrast correction)
Inter-tomogram and intra-class variability : Even within single preprocessing tech- nique, there is substantial variability between different tomograms due varying acquisi- tion conditions (c.g., sample thickness, ice quality), causing appearance changes for parti- cles of the same class (see Fig. 7
In Supp. Table 18, we quantitatively summarize the voxel intensity distributions, confirming these significant photometric shifts. For example, the minimum intensity for virus-like particle can vary by more than a factor of two between the training set (-15.78 10-5) and the validation set (-7.08 x 10 10-3), while the median intensity for a ribosome even changes sign between training (0.06 x 10-5) 5) and testing (-0.01 10 10-5). This complex, multi-level variation creates realistic and challenging testbed for
generalization.
Data Preprocessing and Augmentation. During training, we only apply geometric data augmen- lation to enhance model generalization. This includes: (1) random flips along each of the three spatial axes (depth, height, width), each with 50% probability, and (2) random rotations of 0, 90, 180, or 270 degrees in the height-width plane.
Model Architectures. For this task, we compare a Standard 3D ResNet with SEg, SEqSI, Affeq variant. All models are based on a 3D ResNet architecture inspired by ResNet-18 (He et al. 2015). adapted for 64 x 64 x 64 input patches. They share the same overall structure but differ in their spe- cific layer configurations to reflect their respective properties. To preserve its theoretical guarantees, the the SEqSI and AffEq models use reflection padding, whereas the Standard and SEq models use standard zero-padding.
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OVERALL ARCHITECTURE. The network begins with initial convolutional layer, followed by four stages of residual blocks, and concludes with a classification head. The initial layer is a '3 3 x 3' convolution that maps the single-channel imput to 64 feature maps. For the SEqSI model, this is a shift-nvariant convolution ("InCConvdd') while the Standard model uses a standard convolution followed by a "BatchNorm3d' layer (in the case of Stiq, SEqSI and Affliq no 'Batch- Norm3d' layer were used as it breaks invariant properties). In both cases, a 'LeakyReLU" activation applied. The network then proceeds through four residual stages, each composed of residual blocks. Stage operates on 64-channel feature maps at '64 x 64 x 64' resolution. Stages 2. 3, and double the channel count (to 128, 256, and 512. respectively) while halving the spatial resolution at each step (to '32x32x32', '16x 16x 16" ,nnd '''8x8"). The classification head processes the final 'x X152' feature map with global average pooling layer to produce a 512-dimensional vector. This vector is then projected to 6 output logits by fully connected layer, which includes a bias term for the Standard model but not for the SEq. SEqSI and AffEq models. (for recall, SEq, SE and AffEq do not use any bias term, as it breaks scale equivariant properties)
RESIDUAL BLOCK STRUCTURE. Each residual block contains a main path with two '3x3x3' bias- free convolutions and shorteut connection. The data flow within the main path is 'Conv -> BN (if Standard) + LeakyReLU + Conv + BN (if Standard) The output of this path is added to the shorteut connection's 's output, and final 'LeakyReLU' activation is applied to the sum. When downsampling is required (at the beginning of stages 2, 3, and 4), the shortcut connection uses '3x3x3' convolution with a stride of to match the main path 's output dimensions.
Training Details. All models are trained using the AdamW optimizer with initial learning rate of lc-4 and the cross-entropy loss function. The learning rate is adjusted at cach step using a cosine annealing schedule with minimum rate of 1e-6. The batch size training epochs was adapted to the training data: 300 epochs for the main experiment on the noisy the cleaner Denoised data. All
set to 8. The number of
WBP data, and 200 epochs for the supplementary experiment models were implemented in PyTorch Lightning and trained on a single NVIDIA A40 GPU using 'bfl 'bfl6-mixed' precision. To ensure robust and reproducible results, we repeated each experiment with 5 distinct random seeds (5, 7, 42, 137, 181). For each run, the final model was selected by taking the checkpoint that achieved the highest validation accuracy.
Training on Denoised Data. We conducted supplementary experi- ment where models were trained exclusively on the cleaner Denoised data. The results, presented in Table 20 show that while the generalization gap is smaller when training the SE model still significantly outperforms the Standard baseline on two of the three out-of- distribution domains. This confirms that the choice of input data influences the disti--iistiibuttinn (OOD) capabilities, but the architectural properties of SEqSI provides a practical advantage.
Supplementary Results:
less noisy data,
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Denolsed
Wbp
Isonet
Ctfdeconvolved
(og)
Std. Dev: 8.58e-06
Std. Dev: 5.39e-05
Std. Dev: 4.16e-01
Std. Dev: 3 62e-05
Intonsity
Figure 6: Domain Shift Induced by Tomogram Preprocessing. A representative central slice from single raw tomogram is shown after four distinct preprocessing pipelines. The top row presents the visual appearance, while the bottom row displays the corresponding pixel intensity histograms (log- frequency scale). Each histogram is annotated with the standard deviation of its pixel intensities
(o1). direct measure of contrast, calculated as of E (1, I)2, where is the mean pixel intensity. The stark variations in contrast and pixel distributions illustrate the severe domain shift classification model is exposed motivating the development of robust methods.
Variability of virus like partide' in 'bonetcorrected scross tomograms
TS 6
TS 5
TS 86
TS 6.6
(log)
Std. Dev: 3.13e-01
Std. Dev: 4.32e-01
Std. Dev: 5. 5.79e-01
Std. Dev: 6.56e-01
12>
15
16
155
0.5
Pixe Intensity
Poxel Intensity
Pixel Intensity
Poxel Intensity
Figure 7: Intra-Class Variability within Single Preprocessing Domain. This figure illustrates the appearance diversity for single partiele class (here, tirus-like-particle) extracted from different tomograms, all processed with the same pipeline (isonetcorrected), The samples selected span the full spectrum of observed contrast, from lowest (left) to highest (right). The top row shows a central slice of each particle, while the bottom presents its pixel intensity histogram, annotated with its standard deviation (σ1). This significant intra-class variability, even within a supposedly homogeneous domain, poses fundamental challenge for the generalization of classification models.
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Table 18: Voxel intensity distribution per macromolecule across the training, validation, and test tomograms. All values are scaled by 10 The statisties highlight the significant photometric shifts between the different data splits.
Voxel Intensity (x (x10-9)
Macromolecule
Dataset
Min
Q1 (25%)
Median
Q3 (75%)
Max
'poofferrtin'
Train
14.17
-0.29
0.10
0.45
4.32
Val
-7.79
-0.27
0.08
0.41
3.15
Test
-8.94
-0.33
0.07
0.43
3.63
"heta-amylase'
Train
-9.47
-0.24
0.12
0.45
3.67
Val
-8.21
-0.26
0.09
0,41
2.88
Test
-7.73
-0.29
0.08
0.42
2.99
'beta-galactosidaas""
Train
-11.34
-0.26
0.10
0.44
3.63
Val
-7.35
-0.26
0.10
0.42
3.48
Test
-9.04
-0.26
0.11
0.44
3.63
'ribosome'
Train
-1.04
-0.46
0.06
0.46
3.75
Val
-8.39
-0.48
0.03
0.43
3.48
Test
-9.89
-0.59
-0.01
0.41
3.63
'thyroglobulin"
Train 15.04
-0.25
0.11
0.44
3.84
Val -8.29
-0.27
0.09
0.42
3.15
Test
-9.04
-0.30
0.08
0.42
3.63
mlus-like-particle
Train
-15.78
-0.30
0.09
0.44
3.98
Val -7.08
-0.30
0.07
0.40
3.15
Test
-8.46
-0.42
0.03
0.40
3.29
Performance comparison on CZI test tomograms. Models were trained on WBP data, evaluated over random seeds, for 300 epochs. Results are reported as mean + std. Best results for each metric are in bold.
Table 19:
Standard Acc. (%)
SEqSI Acc. (%)
Data Type
WBP (in-distribution)
87.2 1.7
85.2 4.0
Denoised
22.4 + 7.4
74.5 4.8
IsoNet Corrected
16.0 + 1.6
73.2 + 4.7
CTF Deconvolved
48.9 + 12.3
66.5 + 10.9
Table 20: Performance comparison on CZI test tomograms. Models trained on Denoised data, evaluated with 5 seeds, for 200 epochs. Results are reported as mean + std. Best results for each metric are in bold.
Standard Acc. (%)
SEqSI Acc. (%)
Data Type
Denoised (in-distribution)
88.0 + 1.9
88.6 + 5.0
IsoNet Corrected
16.7 + 0.0
76.7 + 9.7
CTF Deconvolved
16.4 + 0.4
21.9 + 4.7
WBP
17.1 + 1.1
16.9 + 0.4
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D.2 ARCHITECTURAL PROPERTIES VS. NORMALIZATION PREPROCESSING, FOR DOMAIN SHIFT ROBUSTNESS
Motivation. While the min-max normalization provides invariance to global affine photometric transformations B.4 this experiment investigates the limitations of min-max normalization in han- dling domain shifts, contrasting it with the inherent robustness of SEqSI and AffEq architectures.
Experimental Setup. A Standard model trained and test with the min-max normalization is com- pared against the Standard, SEq, AffEq and SEqSI models trained and test without normalization.
Results and Analysis. Table demonstrates that the performance of MinMax normalized Stan- dard model drops significantly on duttrf-distributionn domains (16.98% on IsoNet Corrected and 17.84% on Denoised) compared to 84.81% WBP. This indicates that the min-max normaliza- tion does not ensure generalization across different data distributions.
Conclusion. The experiment highlights the limitations of pre-processing for robustness. Embed- ding invariance and equivariance directly into the network architecture, as with SEgSI and AffEq, is more effective for handling domain shifts, enabling better generalization to unseen data domains.
Performance comparison on test set. Standard baseline is trained with and without the MinMax normalization, during both train and test. Models were trained on WBP data. Results are reported as mean accuracy (Ace.) = std. Best results for each metric are bold.
Table 21:
Train/Test Norm.
MinMax
Standard Ace. (%)
Standard Acc. (%)
SEq Acc. (%)
SEqSI Acc. (%)
AffEq Acc. (%)
Data Type
WBP (in-distrib.)
84.81 + 1.87
87.17 + 1.50
87.69
85.15.13.59
79.26 1: 4.04
CTF Deconvolved
41.34 + 11.20
48.91 + 11.04
6..21
66.51 + 9.75
79.07 + 5.04
Denoised
17.84 + 1.96
22.36 + 6,59
28.42
74.53=t4.25
61.79 + 5.97
IsoNet Corrected
16.98 + 0.63
15.95 + 1.44
16.67-10.00
73.21
46.37 1: 5.98
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E EXPERIMENTS IN OBJECT LOCALIZATION APPENDIX
E.1 DETAILS ON THEORETICAL CONTRIBUTIONS
E.1.1 PROOF OF INVARIANCE FOR OUR SCORE MAP GENERATION APPROACH
For AffEg network:
2 ((f(Tx,,
2(Txx by equivariance properties
\f(x) + p [[Af(x) +
(Af(2)) p))
Af(x) + (B)
by mean and std properties
{x(f(x))
2(S(z)))
For SI network:
(((()x( = (()))(((z)) by equivariance and invariance properties \f(x) - [()
(A)()))
Xf(x) - [x[
by mean and std properties
{a(((=)
2(f())).
E.1.2 CONSISTENCY IN RESULTS REGARDING THE CHOICE OF THE NORMALIZATION POST NETWORK
Common normalization functions include min-max scaling to [0, 1] (T1 x scaling to 1- 1] (T2 + (xx 1), and standardization (T3 +
x-min(x) max(a) minn(a))
a(x)
For the centroid detection task, the strategy we apply imply to normalize the output of the Af- fEq and SEqSI networks. We have decided to apply a standardization strategy which is equiv- alent to function T3. More precisely the standardization of the output corresponds the affine normalization [()a( /// ((+]( denoted his
whose inverse transformation is the normalization
We show that rescaling the outputs using Ti would lead to an equivalent thresholding strategy and an identical proof can be given for T2.
The normalization in [0. 1] using Tj corresponds to the affine normalization (f whose inverse transformation is the normalization 7 [maxx-in))/iees (=)), mmi(f((=) denoted h1
Ti/(max - (enn)//(())).mi(((())///mmm
We consider that network f is applied on an image æ giving y = f(x)).
Let 73 be threshold that separates the regions of interest from the background the normalized output channel the network using T3.
Hence, the threshold 71 = Ti (ha(7a)) provides the exact same separation for the normalization Ti.
(f))))) S 73
+ f(x) < ha(1s) Ti(f(z)) s T1((((7)) Y
This holds because all the normalizations studied are increasing functions.
Experimentally, obtain thresholds by optimizing given metric. So, if "s is the optimal threshold for normalization T3. then 71 is optimal for Ti and that will be the chosen threshold for that normalization.
Moreover, the demonstration provided in Supp. properties instead of mean and std properties.
holds for Ti and Tp using min and max
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E.1.3 INTUITION ON NORMALIZATION BENEFITS IN ALTERATION ROBUSTNESS
We can wonder why our strategy is better than the classical approach we still apply a normaliza- tion after the network and it is only moved from "before the network" to "after the network". In the standard approach, the input data scaled fit a chosen range or distribution, then fed the net- work to provide the output channels and score map. But some photometric alterations can affect the normalization and modify the range of the really meaningful data, leading to wrong prediction from the network as the data is not properly scaled/shifted accordingly to the training data. For example, an image contains very bright artifacts, the normalization specific range will be affected and the rest of the data will not get the same value after normalization that it would have without arti- facts. In our paradigm, the network being invariant or equivariant, the learned features do not rely on the input range and does the output of the network. It is then robust to various transformations that could alter the normalization range the classical approach. This output is then normalized fit desired range distribution for the sake of interpretation, then thresholded. Applying this strategy provides robust results, as well as increased generalization.
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E.2 DETAILS ON EXPERIMENTAL SETUP
E.2.1 EVALUATED ARCHITECTURES
The four variants of U-Net Ronneberger et al. (2015) that consider have similar architecture: the initial number of channels equals to 64 for 2D, 32 for 3D models, maximum depth of 4, convolution with kernel of size 3 3 at each depth, max-pooling dividing the dimension of channel by 2 along each dimension and doubling the number of channels.
The specificity of each model are the following ones:
Standard: our baseline, unconstrained CNN with standard convolutions using bias and ReLU activation functions,
SEq: bias free convolutions with unconstrained kernel weights and ReLU activation func- tions,
AffEq: bias free convolutions with kernel weights constrained to sum to 1 and Sort-pool activation functions,
SEqSI: the first layer of the U-Net generating the 32 channels from the imput image is shift invariant (weights constrained to sum to 0), the rest of the network is composed of unconstrained bias free convolution and ReLU activations.
To implement the models for the object location task, we applied the "telescopic" strategy (see App. to constraint the weights.
E.2.2 OBJECT LOCATION DATASETS
Data Science Bowl 2018 The Data Science Bowl 2018 (DSB) dataset (Goodman et al. 2018 is dataset of 2D real microscopy images from Kaggle challenge. We focused on the subset containing 497 fluorescence images used train StarDist 2D (Schmidt al. 2018). Each image contains dozens of objects. The set is split in 427 training images, 25 validation images, 50 testing images. The test set contain in total approximately 2500 objects representing a wide range of scenario of shape and illumination conditions that can be observed in fluorescence microscopy. Some example are shown Supp. Fig.
3D synthetic dataset Our 3D synthetic dataset composed of 10 training images, validation images and 5 testing images. Each image contains approximately 400 objects which are ellipsoids with random size, orientation, intensity and the possibility to have a darker inside as it can sometimes be observed microscopy image. Each image is blurred and to mimic real imaging conditions (Boulanger et al. 2010). An example of such image is shown in Supp. Fig.
noise was added
E.2.3 DEFINITION THE ACCURACY METRIC
We can compute a distance matrix E IR Nerpred XNpred, that stores the distance between all pairs of GT GT-predicted points: D.,, |C; č 2اارu is the distance between the GT location c, of index and the predicted location é, of index j. The pairing between points is done by iteratively associating together the points that correspond to the minimum of D and removing the corresponding column and row to prevent the process to reassign those points. We also investigated the solution that consists in solving an assignment problem between the points sets that minimize the sum of the distances of all pairings, as it is often done for other tasks such as segmentation. But, in our case, minimizing global term sometimes results in associating point with another that is far away instead of one really close as it minimize the overall process. We do not want this behavior to occur (see Fig. 10) thus keep the greedy pairing approach depicted first. For each matching between predicted and a GT center, we know the distance d in pixels (or voxels in 3D) between the two objects.
In what follows, we define:
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Figure 8: Examples of images from the test set of DSB dataset.
(a) 3D volume from the synthetic dataset
(b) 2D slice extracted from the 3D volume
Figure 9: Illustration of the 3D synthetic dataset mimicking fluorescence microscopy
TP(d) (True Positives): the number matched GT and predicted centers separated by less than d pixels/voxels apart;
(dd (False Positives): the number of predicted centers unmatched a distance greater than or equal to pixels/voxels from their paired GT center;
(d) (False Negatives): the number of GT centers unmatched or at a distance greater than equal to d pixels/voxels from their paired predicted center.
We evaluate the performances of the tested approaches using the accuracy metrie:
TP(d) acc(d) TP(d) FN) + FP(d)
Eventually, our final score is provided by the area under the accuracy curve, integrated between and maximum distance D:
L' acc(d)dd.
score D
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12,05
X GT points
Pred. points
Figure 10:
We illustrate the two pairings strategy between predicted and GT locations. The green pairing minimizes the global cost corresponding to the sum of all pairing distances. However, it does not pair together the two points that right red point is and center points correspond that consider the black pairs. The very high distance pair will be filtered afterward anyway as we only consider pairings above certain distance threshold d to compute the accuracy.
distance of one which is the behavior that expect (the left blue point is FN, the a match). Thus, we prefer greedy pairing approach
In the experiments, choose the value of that we use, we consider that pairing for which the distance between objects is equal to the mean radius of the objects in the set can be considered as wrong matching (the predicted point is probably outside the object or almost outside it). Thus we fix D equal to the mean radius for each set. In the 2D set, the mean radius is equal to 11.99 pixels, we fix the threshold to D = 12. In 3D, the radius is approximately 5.53 voxels, thus we fix the distance threshold to D 6. In the Tables, we report the scores for those D.
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Table 22: Measure of the invariance for the object localization task under various afline in- tensity transformations. The invariance measure is equivalent to computing the accuracy for 1, comparing new locations with reference locations obtained in the original range [0,I]. A value of I corresponds to an invariant location estimation. Results in gray are invariant.
w. sigmoid and BCE SEq
w. standardization and ZMSE
Setting
Standard
AffEq
SEqSI
Standard
SEq
AffEq
SEqSI
-2
0.0
0.0
0.112
1.0
0.0
0.001
1.0
1.0
0.5
0.076
0.037
0.832
1.0
0.085
0.101
1.0
1.0
Shift
μ 2
0.001
0.0
0.74
1.0
0.003
0.002
1.0
1.0
μ 10
0.0
0.0
0.22
0.002
0.0
1.0
1.0
À 0.5
0.264
0.97
0.959
0,984
0.365
1.0
1.0
1.0
Scale
0.157
0.869
0.897
0,927
0,238
1.0
1.0
À 255
0.003
0,014
0.012
0.012
0.032
1.0
1.0
1.0
μ -2, 10
0.029
0.055
0.526
0.309
0.016
0.001
1.0
1.0
Affine
5, À = 0.1
0.0
0.0
0.101
0.953
0.002
0.0
1.0
1.0
5, A
0.001
0.0
0.67
0.927
0.003
0.002
1.0
1.0
E.3 EXPERIMENTAL RESULTS
E.3.1 VALIDATION OF OUR TRAINING AND INFERENCE STRATEGY TO GENERATE INVARIANT CENTROID DETECTION ON DSB2018
This experiment was conducted using 2D U-Net models. We trained the models for 1000 epochs, using batch of size 64, using AdamW optimizer with a learning rate (Ir) of 10 and image patches of size 128 x 128, each patch being ranged in the range [0. 1]. For each approach, we selected the weights that minimized the validation loss during the training to avoid taking model overfitting the training set. We optimized the thresholding value by maximizing the area under the accuracy curve on the validation set.
We perform first experiment to verify empirically the theoretical invariance of our approach. We evaluated models: the architectures Stand., SEq. AffEq and SEqSI using either the standard thresholding and loss (BCE) strategy or approach based on standardization and ZMSE loss. The additional combinations: Stand. +ZMSE, SEq+ AffEq+ + BCE and SEqSI+ BCE were added here to show that neither the network nor the post-processing alone guarantee the invariance and both are necessary.
We evaluate the invariance by first estimating the objects locations for image scaled in [0, 1] range to obtain reference locations. Then, we re-estimate the object positions for different values of scale, shift and affine normalization. We then compute invariance measure, equivalent to the accuracy for d = 1 between the new locations and the locations estimated in [0, 1]. If the measure is not equal 1, it is either because: i) there are missing additional location for that transformation creating either FP or FN; ii) there are the same number of objects which are not at the same positions (as we set d we consider only matching of objects whose positions are distant by less than 1). Thus, an invariant location estimation corresponds to value of 1.
Invariance results are given in Supp. Table 22 invariant when paired with the standardization and 7MSE loss.
As expected, only AffEq and SEqSI are completely
We also provide additional results assessing the performances of each of these methods on this dataset. We assess each of the above mentioned methods in [0, 1] and other ranges that are either scale, shift or both. Supp. Fig. [ shows the performances (accuracy curve) of each method for these different ranges of normalization. On the training range, all approaches provide similar and very satisfying results with very close accuracy curves. However, we observe that, as expected, the non invariant strategy have major drops in performances for other ranges:
Shift (μ 1), both AffEq+ZMSE and SEqSI+ZMSE provide invariant results. SEqSI+ provides invariant results too as the network shift-invariant by itself and do not rely on the post-processing remove the shift. Surprinsingly, AffEq+ BCE shows good performances
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0.9
0.9
0.8
0.8
ordinary_BBl BCE
Accuracy
SEq_BCE
Accuracy
0.6
0.6
AffEq_BCE
0.5
SEqSI_CCE
ordinary_ ZMSE
0.4
0.4
SEq_ZMSE
0.3
AffEq_ ZMSE
SEqSI_ZMSE
0.1
10
10
12
Distance to GT (in pixel)
Distance to GT (in pixel)
(a) Range [0, 1]
(b) Shift μ
0.8
Accuracy
Accuracy
0.4
0.4
0.2
0.0
10
12
Distance to GT (in pixel)
Distance to GT (in pixel)
(c) Scale À = 0.2
(d) Affine À = 2, = -1
Figure 11: Accuracy curves for various range of normalization evaluated on DSB2018 test set. The range used to train the networks was [0, 1].
too, but it cannot be explained by any invariant property. The performances of the other methods drop.
Scale (A 0.2), AMSE, SEqSI+ZMSE and SEq+ZMSE provide invariant results as the three networks are scale-equivariant and the thresholding strategy makes the location invariant. Other methods performances drop.
Affine (A = 2. = 1), AffEq+ and SEqSI+7MSE provide invariant results. Aʃ- BCC and SEG provide non-invariant, but still good results. We explain this by the fact that even the thresholding strategy does not provide invariant results, the output of the networks being equivariant, the maps conserve the pixel values order. Thus, if the thresholding is not too sensitive to the chosen threshold (for maps that provide very different values for background and foreground, different threshold could provide similar results) and the network output not shift in the sigmoid tails (where it is almost 0 or I ev- erywhere), the location can still be accurate even if it not invariant. But this results could not be generalize to any affine transformation. All other methods show important accuracy drops.
On Supp. Figures 12 we present the output score maps and predicted locations for each method, on the same image, for the normalization range [-1, 1]. These results illustrate how the non-invariant methods score maps can be affected by a change in the input normalization range. Obviously, this perturbations of the score map significantly affects the object localization.
In general the normalization range is the same during training and inference and that experiment was mainly dedicated empirically prove theoretical results. In the following experiment,
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Image
GT locations
Pred. Score Map
Pred. Locations
Score Map
Pred. Locations
Standard+Bor BCE
AffEq+BCE
Standard ZMSE
AffE
SEQ+BCE
SEQSI+BCE
ZMS+
ZMQE
Figure 12: Score maps and object location for the different methods in the range [-1, 1]. The GT location are represented with the large green dots and the predicted location are represented using the smaller red dots.
evaluate if, for the same range of normalization, invariant strategies can improve the robustness over non-affine perturbation.
E.3.2 ROBUSTNESS OF Standard WITH NORMALIZATION AND SEqSI AGAINST OPATIALLY-VARYIIE SHIFTS
In the Methods Section (see Section we present our approach that makes a convolutional net- work provably invariant global affne photometric transformation applied to the image. Other widespread methods lead to similar results. Most of the normalizations commonly applied in im- age processing eliminate global affine photometric corruptions. For example, the transformation z-min(æ)
T x + max(a))-mi(()) which normalizes the image to the range [0,1], removes the shift by subtracting min(x:) and the scale by dividing by max(æ)) min(x:). Similarly, the normalization
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Ta x 1 2- --B[a] global shift and scale.
[[a] which centers and reduces the intensity distribution of the image, also eliminates
Thus, at first glance, one might therefore think that our approach has little value if normalizing input images provides a simple solution the same problem.
In this section, we demonstrate how our approach differs from simple normalization, particularly for images affected by affine intensity transformations that vary spatially (cases for which theoretical guarantees are weakened but, in our view, still meaningful). We describe two examples of local intensity shifts for which our approach theoretically guarantees better results than simple normal- ization, and we illustrate it with experimental results.
For the first example, we consider the very simple case of two-piece shift μr: the upper half of the image undergoes no transformation, while the lower half undergoes a shift. Specifically, considering the domain of positions on the image defined as [0, H 1] x [o, W - 1] for an image of size (H, W), v(i.j) E [0, - 1] [o, W W-1,, Mi.j Of I H/2 and Mid
if >> H/2. In this configuration, applying the transformation Ti to rescale the original image to [0.1] will, at best, behave adequately on one half of the image (depending were located before and after the transformation) and, at worst, behave poorly everywhere (meaning the normalized image values will differ from those of the original image at every location). Similar considerations apply if T2 is used.
where the minimum and maximum values of the image
In the case of the SEqSI network, the shift is removed by the first layer of the network through convolution that acts locally. Thus, the shift is correctly eliminated almost everywhere, except at the shift discontinuity. The size of the affected area after the first layer depends on the size of the convolution kernel. In our case (3 x 3), only two rows of pixels are impacted. Depending on the number of subsequent convolutions performed by the network, the error will propagate a slightly larger region, but significant portion of the image will remain unaffected by this non- non-global shift. In light of this observation, we can note that our network provides guarantees of near-invariance to piecewise constant shifts.
To illustrate this observation, we present in Supp Fig. 13 the object localization results for the case where the image is corrupted with a shift of μu 0.25 applied only to the lower half of the image. We evaluate the Standard network from the experiment in Section E.3.1 and we apply a normalization to [0.1] (namely Ti) as pre-processing to match the range used during training. We compare with the SEqSI network from the same experiment.
We can clearly observe that, for the Standard network, the score map is significantly affected across the entire lower half, even far from the junction. In contrast, the score map for SEqSI appears to show differences only in the immediate vicinity of the separation. Regarding position predictions, that SEqSI produces very similar position predictions between the clean image and the corrupted image. Some positions are missing, particularly because the thresholding depends on global values of the score map (mean and standard deviation). which are inevitably affected by the differences at the junction. On the other hand, Standard misses large portion of the centers in the lower half, and those that are predicted are generally slightly displaced compared to the positions predicted on the clean image. This difference in behavior explains the reasonable invariance measurement for SEqSI, whereas that of Standard is very low.
note
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Images predicted
Predicted maps
Original image
Standard with normalization [0,1]
Shift of 0.25 on lower half
Invariance measure: 0.081
Original image
SEqSI
Shift of 0.25 on lower half
Invariance measure: 0.416
Figure 13: Object localization results for spatially-varying shift corresponding shift of μ 0.25 only on the second half of the image. The red dots correspond to the positions estimated on the clean image, the green one are obtained on the corrupted one. On the corrupted image. both red and green are displayed: the non-visible red correspond to green that perfectly overlap (invariant estimation). The "invariance measure" correspond to accuracy for comparing green and red positions (the closer is the invariant estimation is).
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The second example consider is that of a shift that evolves linearly according to the position the image. Specifically, we define a shift il such that v(i.j) E [0. - 1] x [0, W 1], Pi.j = Re M(## ) where K is parameter controlling the magnitude of the shift. This kind of shift can be observed frequently in biological images, for example where the thickness of the imaged tissue is not constant.
Just in the previous experiment, basic normalization [0.1] will not achieve the desired behavior (removing the shift) anywhere.
For our approach, since the shift is not piecewise constant, the argument put forward for the previous experiment no longer holds. However, we can examine the form of the shift in the neighborhood of position (i) on the image. For (.+1..) the shift is (l++,, = Pi.j + H For any other pixel (k: 1) in the neighborhood, the shift can also be decomposed into the sum of common component Ma.j de- pending on (i,j) and a residual component d;-kj-r depending on the location in the neighborhood: Mi, + d Thus, the convolution (which sums zero over the pixels in the neighborhood) will eliminate the common component Pi,y related to the position on the image and will have an indeterminate effect on the residual part. In cases where the residual part is negligible, particularly when K. is small compared to the dimensions of the image (or when the shift varies slowly a more general, non synthetic setting), the proposed approach therefore appropriately removes most of the shift.
In Figure 14 illustrate the localization results for the Standard network using [0.1] normalization as pre- pre-processing. alongside those provided by our SEqSI, for spatially-varying shift with x 0.3.
The results are more striking for this experiment: the score map of the Standard network is heavily impacted, especially in the bottom-right corner where the shift is strongest. As result, position prediction no longer works at all, and the invariance measure is almost zero. For SEqSI, the network predicts nearly identical results, explaining the very high invariance measure.
more real life example of apatially-varying shifts that are handled very well by our approach is the bright-artifact case mentioned at the end of Section (7.2 and more developed in Supp. E.3.4] In that case we generated very local and high intensity shift applied in small regions of the image to generate artifacts that can be observed in microscopy images.
In light of the various points discussed this section, it seems to us that, although our solution may appear simple at first glance, it addresses a much broader range of corruptions than most of the normalizations commonly used deep learning tasks. Exploring its applicability across multiple applications by testing its robustness to numerous non-affine and/or non-global corruptions seems highly relevant.
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Images predicted centers
Predicted maps
Original image
Standard with normalization [0,1]
Shift linearly evol. with position
Invariance measure: 0,010
Original image
SEqSI
Shift linearly evol. with position
Invariance measure: 0.719
Figure 14: Object localization results for spatially- shift with 0.3. The red dots correspond to the positions estimated the clean image, the green one obtained the corrupted one. On the corrupted image, hoth red and green are displayed: the non-visible variant estimation). The "invariance measure" correspond to an accuracy for d = comparing green and red positions (the closer it is to the more invariant the estimation is).
correspond to green that perfectly overlap (in-
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(a) 2D slice
(b) 3D volume
Figure 15: Example bright artifacts (red arrows) on 3D biological image representing zebra fish larva. The maximum contrast has been limited to allow visualization because the artifact is 250 times brighter than other pixel values.
E.3.3 ROBUSTNESS TO AFFINE AND NON-AFFINE PHOTOMETRIC CORRUPTIONS
This experiment was conducted using 3D U-Net models. We trained them for 6000 epochs, using batch of size 4, using AdamW optimizer with a learning rate (Ir) of 10 and image patches of size 90 x 90 x 90 voxels. Each patch is ranged in the range [0, 1].
If data augmentation strategy applied (Aff., NAfF. or All, details Supp. Sec. C.2), the patch is corrupted by image transformation picked in that augmentation category. The parameters of the corruption are picked in the same interval than those presented in the classification experiment.
Gradient clipping is applied to prevent exploding gradient. We observe that the AffEq method is particularly hard to train in 3D for this task and often creates exploding gradients. Clipping partially solved the problem but this tendency to create high gradient can explain the bad results of AffEq, especially when combined with data augmentations that perturbs the training even more.
For every approach, selected the weights that minimized the validation loss during the training to avoid taking an over-fitted model.
We optimized the thresholding value used in the local maxima detection by maximizing the score on the validation set.
To assess the robustness random alteration picked for each image of the set (with parameters picked in the same interval than for training). We fixed the random seeds guarantee that each model sees identical transformations.
each perturbation category, we inferred the models the with
For each experiment and each model, we launched 5 training using 5 different seeds (f1, 2, 3, 4, 5)) ensure statistical significance of the results. We provide results corresponding the mean of the 5 models score. They are provided with standard deviations in Supp. Table 23
E.3.4 ROBUSTNESS TO BRIGHT ARTIFACTS
Bright artifacts, caused by sensor saturation arise frequently on fluorescence microscopy images. They correspond to very bright zones in the image that are sometimes thousands times higher in intensity than the maximum value of the signal (see example in Supp- Fig. L5p. When the image is normalized in particular range (e.g. j0. 1]) without particular concern on corrupted zones, the rest of the signal is squeezed. Thus, rescaling the image [0, 1]. locally correspond to scaling compressing the signal in parts of the images with artifact.
Therefore, we assume that even if these alterations are neither affine nor global, our invariant ap- proach handle them correctly, given that a convolutional network processes an image locally and that the score maps are voxel results. In presence artifacts, the overall result will not be invariant expected: i) the results will differ at locations corresponding to artifacts voxels, ii) the mean and std of the predicted output will be impacted, modifying the score map after standardiza- tion. But if the artifacts represent small region of the image, the mean and std could vary slightly, still providing a good localization even without any data-augmentation.
To evaluate this hypothesis, corrupt test images with artifacts with various intensities:
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Table 23: Robustness to perturbations at inference on the 3D synthetic test of fluorescence microscopy (mean-istd over 5 models), Test score for D = 6 of four architectures under various photometric corruptions. Models are grouped by the training augmentation strategy used: none (0), affine (Aff.), non-affine (NAff.), or all combined (All). Fach row corresponds to a different perturbation applied at evaluation. For each combi- nation of mode++augmentation strategy, models were trained, we give the mean score and the std computed among 5 models. Within each training strategy (column group), the best result is highlighted in gray. The overall best score for each perturbation is in bold.
Train Aug. = 1
Train Aug. Aff
Model
Corruptiont
Stand.
SEq
SEqSI
AtlE
Stand.
SEq
SEqSI
Afl
Original
0.868-0.015
0.005+0.33
0
0.87+00.007
0.868-00.005
0.842+0.011
0.881-0.000
0.395++0.017
Affine tramsformations (AIL)
Shift
0,152:0.12
0.11740.135
0.886=00.002
0.87:00.007
0.864=0.005
0.83740.016
0.881=0.006
0.395
Seale (<. 1)
0.12-00177
0.558+0.319
0.886-0.002
0.87+0.0077
0.866+00.005
0 153
-
0.395++0.017
Scale (>x 1)
0.491=0,142
0.50612.287
0.886:0.0022
0.87:10.007
0.861:200005
0.835=00.22
0.881=0.006
0.395
Affine
0,149=0,109
0.093:00.07
0.886=40002
0.87:00.007
0.843=00.028
0,799:0.066
0.881=0.006
0,395+00.17
Non-Afline transformations (NAff.)
Shift saturated
0.074-0.073
0.066+00.077
0.069-0.059
0.389+0.179 179
0
0.298+0.22
0.322+00.1
0.264+0.06
Scale (> 1) saturated
0.51=0.136
0.444:0.272
0.576:0:0088
0.842:0.019
0.84:000166
0,792:0.014
0:846:00011
0,299+0.036
Affine saturated
0.124=0.069
0.09410.063
0.11320.003
0:45610.1
0.568
0.374:1.117
0.398_0.179
0,297.10.041
Noise low
0.633-0.122
0.505+0.229
0.885=0.000
0.736+00:1 122
0.841+00013
0.882-0.006
0.385+0.018
Noise high
0.233=0.14]
0,045:0005
0.08:=0.077
0.238 127
0.603 236
0.42+00186
0.08=0.117
0,102=00.052
Gamma (darken)
0.535_0.136
0.16210.214
0.42420.198
0.70110.00
0.838.0.015
0.67610.104
0.669-0.175
0.38810.017
Garma (ighten)
0.512:20.23
0,3910.278
0.883=0.002
0.805
0.865=000004
0.835:0.012
0.876=000007
0.38220.02
Additional experiment on artifacts
Arti, low
0.561=0.165
0,571#0
0:842:0.016
0 008
0,714+0.1344 134
0.87-00009
0
Arti, medium
0.271=00119
0.53:10.342
0.81120.055
0.845:10.007
0.852=00.011
0,48:103388
0,869-0.099
0.37610.023
Arti, high
0.064=00.99
0.081:40.066
0.691=0.146
0.733:00.046
0.846::0.017
0,293:40229
0.863-:0.01
0,291400054
Train Aug. NAIf
Train Aug.
Corruption
Model
Stand.
SEq
SEgSI
AAEq
Stand.
SEq
SEyST
AnEq
Original
0.847=0.051
0.702110221
0.083
0.483:10057
0.877=00.044
0,85310.013
0.881=0.001
0.377:10.006
Affine transformations (Aff.)
Shift
0,599:=0:25
0.289+0.11 168
0.883500003
0,483++0.07
0.874=00.008
0,2
08881 0.0044
0.377+0.006
Scale (<: 1)
0.796=0.125
0.5910.281
0.883-0.003
0.183:10037
0.87120.006
0.78910.081
0.881=0.001
0.377.10.006
Scale (> 1)
0.835_0.067
0.723:00217
0.883=0.003
0.483:60.037
0.875:000055
0.855:10.01
0.377:10.005 0.881=0.004
Affine
0.661==0.00
0.34+0.133
0.883-0.003
0,483+0.057
0.837=000055
0,803-+0.076
0.881 00.004
0.377++0.00
Non-Affine transformations (NAfI.
Shift saturated
0.508-0.115 1333
0.385+0.110 118
0:564=0.199
0.234++0.10 104
0.567 00161
0.405:40.212
0.555=0.162
0.244+00053
Scale (> 1) saturated
0.83=0.069
0.716.10118.
0
0.768:10.073
0.87=00.09
0.823:0:003
0.078=00005
0.284:10.029
Affine saturnted
0.596-0..94
0.423++0:42
0.607+0.1
0.903+-0.049
0.639
0.44++0 226
00.097
0.287++0.033
Noise low
0.846=0.055
0.703:10.21
0.884=0.004
0.471:000366
0.878=0.004
0.85510.012
0:881=0.004
0.372+0.01
Noise high
0.79-0.085
0.62:10.24
0.73510.162
0.074.10.036
0.84410.009
0.763.100066
0.78-0.086
0.08.10.07
Gamma (darken)
0.758 0.127
0.581+00312
0.364=0.018
0.438+0.03
0.873-0.005
0,752+0.067
0.777-0.021
0.376+0.01
Gamma (lghhen)
0.843=0.054
0.71810.199 199
0.884-0.003
0.471:000057
0.876:0.0044
0.85:00013
0:879-0.003
0.364+20013
Additional experiment on artifacts
Ari, low
0.816-0.078
0.569+DD364
0186-0.026
0,477+0.0036
0.862-0.0004
0,81900.022
1L872-0.004
0.374+0.06
Arti medium
0.777-0.100
0.521++0358
0.787=0.148
0.472:40.034
0.859=00.006
0.789+0.064
0.868-0.00
0.364+0011
Arti, high
0.672=0.181
0,343:t0.188
0.757=0.144
0,426:20.015
0.855
0.618:+0.128 128
0.22=00.044
0,273+0.059
low:
the artifacts maximum intensity is 3 times higher than the original signal maximum intensity,
medium:
the artifacts maximum intensity, is 5 times higher than the original signal maxi- mum intensity
high: the artifacts maximum intensity is 10 times higher than the original signal maximum intensity.
We illustrate the corrpptions in Supp. Fig. 16
We evaluate the models trained for the precedent experiment. The images are rescaled in the [o, 1] range before being provided to the network to match the training range. The results are given in the last rows of Supp. Table 23 show that even with no augmentation. both AffEq and SEqSI provided extremely good results for artifacts up to 10 times brighter than the original signal while other methods score is close to O. It shows that our approach addresses this type of corruption intrinsically. Adding data-augmentation improves even more SE performances.
58

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Published as a conference paper at ICLR 2026
With low artifacts
With med. artifacts
With high artifacts
Clean image
Figure 16: 3D volume from our test set of synthetic images corrupted with artifacts with various intensities: 3x the maximum images intensity for low, 5x for medium and 10x for high, respectively.
The common strategy to address bright artifacts, based on normalization by p-value requires to know approximately the amount of pixels/voxels curated by artifacts while we propose approach that automatically address this issue.
F STUDY OF THE BINARY SEGMENTATION TASK
We add final experiment the binary segmentation task, that can be considered sub-task of the object localization (as the later include thresholding of a score map).
F.1 DETAILS ON EXPERIMENTAL SETUP
Evaluated Architectures The architectures are rigorously the same than for the object localiza- tion task (see Supp. Sec. E.2.1 for details).
Binary Segmentation Datasets. The DSB 2018 set mentioned in Supp. Section F.2.2 provides instance segmentation masks that can be converted into binary segmentation masks, thus we decided reuse this for this task.
Definition of the score. For this task, compare the GT binary mask BeT with the predicted binary mask Bpred using a Dice Score (DS).
DS 2 BoT & Bpeed
(16)
Barl + Bpreet]
where @ an element-wise product the masks and |GT| count the number values equal in Ber.
Training details. Unlike the object localization task for which we generated dedicated score maps to train the models (with local maxima at the barycenter of each object), for binary segmentation the GT binary masks can directly be used as GT for the loss computation.
We trained the 2D U-Net models for 1500 epochs using a learning rate of 10-4, using the AdamW optimizer. The batch size is 32, the image patch are of size 128x 128 pixels.
Again, the thresholding value used to generate the binary masks is optimized for each model at the end of the training to maximize the DS over the validation set.
In Table 24
we report the Dice Scores on the DSB test set for each the methods (averaged over models trained for different seeds). In Fig. 17 we show examples of binary masks obtained two images either on original or corrupted images. We observe that the shift makes the baselines either predict full foreground or full background while AffEq and SEqSI results remain unchanged.
On the quantitative results, we observe that, once again, AffEa and SEqSI are affine-invariant, For the experiment with data-augmentation both models demonstrate better robustness than baseline
59

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Published as a conference paper at ICLR 2026
Table 24: Robustness to perturbations inference on the DSB dataset for binary segmentation. The Dice score is given comparing predicted binary mask with GT one. Models are grouped by the training augmentation strategy used: none (0), affine (Aff.,, non-aftine (NAff.), or all combined (All). Each row corresponds to different perturbation applied during evaluation. Within each training strategy (column group), the hest result highlighted in gray. The overall best models trained on different seeds. The std. are given in Supp. Table 25
for each perturbation is in bold. Results are averaged
Train Aug.
1
Train
Alf
Train Aug.
NAIT
Train Aug.
All
Corruption
Model
Stand.
SBq
SBgS1
AlLE
Stadd.
SEq
SEgSI
Stand.
SEA
SE4S1
AnEg
Stand
SEg
SEgS1
AlIa
Original
0,33
0.933
0.915
0.927
0.31
0.909
0.926
0.926
0:932
0.928
0.923
0.919
0.930
0.931
0.924
0,913
Affine transformations (AIT.)
Shift
0.339
0.289
0.925
03927
I1931
0.929
0.926
0.926
0.634
0.442
01925
0.919
10:83
11929
0.924
0.913
Stall (c.1) 13
0.783
0.903
0.935
0.907
B.929
0.902
0.936
0.926
0.914
0.900
0.923
0.019
0.927
0.918
0.924
0.911
Stall (x 13
0.917
[0934
0.925
0.922
0.931
0.999
0.936
0.926
0.893
10933
0.923
0.919
0.933
0.993
0.924
0.913
Alfine
0.366
0.355
0.925
0.927
1.928
0.917
0.926
0.926
0.611
0.449
01025
0.919
0.19
0.22
0.24
0.913
Non- Affine transformatiions (NAFC)
Shift saturned
0.358
0.353
0.536
0.520
0.517
0.532
0.53
0.531
0.672
0.67
0.66
0.64
0.699
0.577
0.657
0.639
Scale (> 1) satunted
0.915
[0925
0.911
0.918
0.92
0.921
03117
0.915
0.231
0.532
0.923
0.199
0.93
0.93
0.923
0.914
Affine sonurated
0.394
0.386
0.587
0.587
0.575
0.5%
0.585
0.583
0.706
0.708
0.697
0.691
0.700
0.655
0.894
0.68
Noisz low
0.928
0.918
0.651
0.913
0.020
0.907
0.827
0.021
0.331
0.93
0.923
0.022
0.920
0.926
0.93
0.15
Noise ligh
01.6
0.489
0.146
0.715
0.706
747
0.15
0.746
0.894
0.894
0.584
0.581
0.87
0.861
0.77
0.822
Camma idarken)
0.747
0.785
0.815
0.8
03877
0.505
0.811
0.806
0.913
0.893
0.589
0.567
0.916
0.884
0.875
0,854
Gamma lighton)
0.875
0.867
0.656
0.834
0.91
0.904
0.856
0.891
0.927
0.938
0.916
0.917
0.927
0.924
0.916
0,908
Image
Standard
SEq
AffEq
SEqSI
Oliginal Images
Shifted images
Figure 17:
Binary segmentation results for different models trained without data-angmentation. The top images correspond to binary segmentation on original images. The bottom examples correspond to images corrupted with a shift.
models, for multiple non-affine perturbations. Using data-augmentation increases the perfor- mance of all models on non-afline corruptions, with competitive and comparable results.
60

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Published as a conference paper at ICLR 2026
Table 25: Robustness to perturbations at inference the DSB dataset for binary segmentation (mean-tstd over 5 models). The Dice are grouped by the training augmentation strategy used: none (0). afline (Aff.), non-affine (NAIf.), or com- bined (All). Each row corresponds to a different perturbation applied during evaluation. Within each training strategy (column group). the best result is highlighted in gray. The overall best score for each perturbation is in bold.
given comparing predicted hinary mask with GT one. Models
Train Aug. = (
Train Aug. Aff
Model
Corruption
SEq
SEgSI
AfT2
SEq
SEgST
AffEg
Original
0.903-0.002
0.93520.002
0.925=00.002
0.927.10.002
0.931
0.929110001
0.926=0.002
0.926:10.001
Affine transformations (Aft.)
Shift
0.339-0.035
0.28910.096
0.22=00.002
0.927=00002
0.931=0.002
0,929
0.926=0.002
0.926+0.001
Scale (< 1)
0.783-0.057
0.903:10.021
0.92510.002
0.927.00.022
0.929-00002
0.922:100088
0.926=000022
0.926.10.001
Scale
0.917-0.006
0.934:0.002
0.925=0.002
0.927=0.0012
0.931=00002
0.929+0.001
0.926=0.002
0.926++0.001
Affine
0.366=0.059
0.35340.091
0.925=0.002
0.927:00002
0.928=0.004
0,917:000115
0.926:0.002
0.926+0.001
Non-Aflne transformations (NAff.)
Shift saturated
0.358=0.061
0.35340.009
0.52610.041
0.52910.019
0.517=0.052
0.52210.051
0.531
0.531100009
Scale (> 1) saturated
0.918:00004
0917=00003
Affine saturated
0.394-0.095
0.386200004
0.307-0.22
0.387=0.029
0.75=00.025
0.378+00022
0.585-0.026
0
Noise low
0.92800:001
0.918110.006
0.851
0.92310.003
0.922:0.001
0.927400002
0.827=0.015
0.92310.001
high
0.746100043
Gamma (darken)
0.747-0.032
0.785+D0017
0811=00.019
0.8+0.0244
00847000000
0.803+0022
0881 0811-0.024
0.806+00.007
Gamma (lighten)
0.875=0.025
0.867:00046
0.866:0.025
0.884220011
0.91:00.007
0,901100007
0.85=00.511
0.89120.02
Train Aug, NAIf
Train Aug.
Model
Corruption
Stand.
SEq
SEqSI
Stand.
SEq
SEqSI
AflEq
Original
0.000
0.928+D.007
0.23+-0.022
0.919+0..022
0.932=00001
0.931+00008
0.924-0.002
0.913++0.066
Affine tramsformations (AIL)
Shift
0.67--0.167
0.44240.049
0.23:10.002
0,919:100022
0.93
0.929100004
0.924-0.002
0.91310.006
Scale (< 1)
0914-00011
0.902+00.02
0923-0.002
0.919+0.002
0.927-00.07
0,918+00017
0.924-0.022
0.913+D0006
Scale (>
0.893-0.002
0.935200001
0.928=0.0022
0.919:10.002
0.932=00.00
0.932200002
0.924=0.002
0.913:20.006
Affine
0611=0.117
0.449:0.072
0.23=00:022
0.919:0.0022
0.919=00.114
0.922:00011
0.924=0.002
0.933+0.006
Non-Affine transformaaions (NAff.)
Shift cnturated
0.672-0.035 0.038
0.67+00009
0.66-00037
0.654+0.039
0
0.657+00044
0.657-0.042
0.639+0.099
Scale (> 1) saturated
0.931=0.00]
0.93250.001
0.92==00002
0.919=0.003
0.93=00.001
0.93:20:00
0.923=0.002
0.91420.004
saturated
0.705-0.027
0.70210.022
0.69720.027
0.69110.03
0.702-0.025
0.694_0.025
0.6810.022
Noise low
0.931 0.002
0.93+40.001
0.23=00.02
0.922+0.001
0.929=000002
0.926:+0.006
0.923-0.003
0.915+0.006
Noise high
0.894-0.006
0.894
0.884=00005
0.881+0.008
0.878=00018
0.861:0.031
0.87=00.011
0.852++0.031
(darken)
0.88910.018
0.916-0.005
0.87510.02
0.8541100119
Gamma (lighten)
0.927=0.002
0.92820.004
0.916:0.005
0.917:200033
0.927=00.002
0.924:100003
0.916:0.005
0.908:20006
61


Reviews:
[{"review_id": "yV63kfgJxx", "reviewer": "Reviewer_FUb7", "summary": "This is a paper which builds on the concept of scale equivariant (SEq) networks, adding shift invariance (SI). This leads to a network which is equivariant to scale operations on pixels in the image, and invariant to shifts in pixel values. In comparison to existing approaches for full affine equivariance, this approach is faster; centrally because it can leverage standard ReLU activations as opposed to the SortPool activations required for full equivariance.\n\nThe authors also propose an approach to preserve the logit outputs of their networks through various post-processing stages such as the argmax operation and thresholding.\n\nThe authors validate their approach for three tasks including classification and localisation. In all cases, the approach demonstrates strong overall performance combined with robustness both to affine and non-affine corruptions of the images.", "questions": "1. In table 5, your SEqSI network outperforms a standard network both in the no augmentation at train/no augmentation at test, and in the affine at train / affine at test setting. Do you have a good heuristic for why this is the case? Is it because of the network's ability to efficiently learn over variations in the train set? It seems to be a curious result to me if the SEqSI approach is missing both layer biases and has a constraint on the first set of weights. \n\n2. In figure 10, the selected values of $\\mu$ and $\\lambda$ are significant in comparison to the range of the original image. As I read it, the chosen value of $\\mu$ would mean there was no overlap between the two distributions in the shift case. Is there a reason that such a significant perturbation was chosen?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper is well structured. The mathematics are clearly presented and easy to follow. The experiments are generally well motivated to demonstrate the benefits which the authors claim to achieve.\n\nThe results which are presented seem to me to be strong. The authors demonstrate the utility of their approach in a controlled setting (CIFAR-10), a macromolecule classification task and a localisation test. In all cases, the SEqSI framework outperforms performs either competitively or outperforms comparisons without degraded performance in the case of unwarped inputs compared to standard network architectures. \n\nIn particular, experiments where networks are compared with various different transforms at train time, and are evaluated under various shifts at test time, are particularly compelling for this architecture.", "weaknesses": "My central reservation about this paper is the magnitude of its technical contribution. In terms of the technical innovative step, it appears to me that the contribution is principally a zero-sum weight in the first layer of the neural network, and a different transformation of output logits to preserve the equivariance/invariance at output.\n\nNonetheless, I would support an acceptance of this paper owing to the results which support the view that the method has clear utility. My view would be strengthened if the authors were able demonstrate the effectiveness of their approach more extensively. While the authors do propose several test settings, and the results appear strong, my concerns are as follows:\n- CIFAR-10 results appear good, and I would particularly credit the wide range of augmentations used within the evaluation framework. However, this dataset is relatively small and not so complicated, and so these results need to be supported by other results in the paper.\n- Macromolecule classification results also appear strong, and this test setting is the most interesting in my view. However, I see that the Affine equivariant and SEq approaches are omitted here, and I cannot find an explanation for this in the paper. I feel that either the inclusion of these baselines, or provision of a reason for their exclusion is essential.\n- The 3D localisation test set is also compelling, and the testing framework is strong here. However, as a synthetic dataset it would be useful to have a real-world comparison. This is presented in the form of the Data Science Bowl set (DSB). But in the main paper the DSB is only mentioned as demonstrating the invariance of the approach, whereas I feel that highlighting the robustness of the method to augmentation (Figure 10) was the most important outcome of this test for supporting the method proposed. I feel that if the DSB dataset is mentioned in the main paper, then the paper would be strengthened by including a table or chart of the main results from that dataset in the main paper (i.e. from Figure 10). Alternatively, I feel that demonstrating utility for a different real-world dataset for another task such as segmentation would strengthen the case for this method significantly.\n\nI also found the presentation of the DSB results in the supplementary results to be confusing in some places. In particular I would highlight that the specification that $d=0$ seems to clash with the definition of TP(d) as \"the number of pairs of centers *less than* d voxels apart\" (emphasis added). I also found the term 'accuracy' in table 16 led me initially to think that this was showing a comparison with ground truth.", "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": "lvhQkZkDWS", "reviewer": "Reviewer_tYPp", "summary": "The paper presents a neural network architecture intended to be “scale-equivariant and shift-invariant” to photometric corruption, but the equivariance and invariance operate only on global pixel-intensity transformations rather than spatial changes. The method prepends a single convolutional layer with zero-sum weights and uses bias-free layers to enforce these properties, and then applies simple post-processing (e.g., standardization) to obtain affine-invariant predictions. The paper evaluates the approach on CIFAR-10, Cryo-ET classification, and microscopy localization, claiming improved robustness.", "questions": "Please see the above comments. My primary concern is the significance, novelty, and timeliness of the work. I would be interested to hear how the authors can clarify the architectural novelty (given that it appears to be a trivial modification) in a way that changes my assessment.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The paper is reasonably well written and clearly structured.\n2. The authors conduct a large number of experiments across several tasks and datasets, providing extensive empirical evaluations.", "weaknesses": "1. The paper initially gives the impression that it addresses spatial affine transformations, but the invariance is only with respect to global intensity changes. Even spatial affine equivariance is already well studied and not particularly novel; restricting the scope further to global intensity scaling and shifting makes the contribution very limited. These transformations are trivial to handle with standard preprocessing or normalization. Removing biases to enforce scale equivariance, or imposing zero-sum weights in the first layer to enforce shift invariance, is not a substantial architectural idea and does not constitute a meaningful contribution.\n2. Despite the many experiments, the paper does not seem to have compare against simple preprocessing baselines such as per-image normalization or mean subtraction. These standard techniques may provide the same robustness without any architectural modification. Without such comparisons, it is unclear that the proposed method offers any practical advantage. In fact, enforcing zero-sum filters in the initial layer appears effectively equivalent to preprocessing.\n3. Overall, the contribution is very limited. The architectural modification is minimal and conceptually straightforward, and robustness to global intensity transforms can already be achieved with standard pipelines. The work does not demonstrate sufficient novelty or significance.", "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": "c3wAg8EaFd", "reviewer": "Reviewer_Ssow", "summary": "The paper introduces SEqSI, a convolutional network that guarantees invariance to global shifts and equivariance to global scales by design. The proposed SEqSI block ensures equivariance to scale and invariance to shifts by constraining the weights of the kernel to be zero-sum in the first block. Equivariance to scale is maintained through the rest of the network via bias-free networks. Notably, the network is significantly less constrained compared to [Herbreteau, 2023] by only requiring weight constraint at the first layer, instead of throughout the network. This yields a network that is significantly more efficient in terms of training, inference, and memory complexity while maintaining the core functionality and achieving on par or better performance. Furthermore, the paper introduces pipeline to extend the affine-equivariance to thresholding tasks such as object localization. This is achieved by replacing the conventional sigmoidal thresholding with a novel thresholding based on the standard deviation of the entire map, which is provably equivariance, as well as a novel Z-MSE loss function.\n\nOn classification tasks, SEqSI is shown to have better performance than baseline models (SEq, AffEq, etc) when no data augmentation is available and performance on par with baselines under augmentation. SEqSI also achieves strong performance to non-affine transformations that it is not specifically designed for, showcasing flexibility. These result showcases the benefits of designing for invariance and equivariance in a convolutional network. \n\nOn object localization tasks SEqSi significantly outperforms baseline methods, including a straight sweep for affine-transformation, both under and without augmentation. It achieves strong performance as well for non-affine transformations. This can be attributed to its novel thresholding scheme that preserves equivariance and invariance throughout the entire transformation.\n\nOverall SEqSI represents an elegant solution to a problem with clear motivations and practical applications, such as biomedical imaging. My initial recommendation is a 6 (weak accept) which should be easily increased to an 8 (strong accept) with a few clarifications from the authors (please see weaknesses and questions section).", "questions": "1. The authors mentioned that the proposed thresholding scheme can also be extended to tasks such as foreground-background segmentation. Would this hold for tasks requiring additional classes such as semantic segmentation? This may further extend the use case of the proposed method for situations such as autonomous driving where cameras are often subject to significantly interference with light sources such as the sun or metallic reflection.\n2. While not completely related, since 2023 there have been works that deal with perceptual variation in images from the point of view of color equivariance. Specifically [1] first introduced hue-equivariance using Group CNNs [2] by rotating the RGB cube through (1,1,1). [3] achieves this using soft equivariance. [4] extends this idea to cover equivariance to hue, saturation, and luminance. I wonder if it would be worth while to talk about these in related works.\n\n[1] Attila  Lengyel, et al. \"Color Equivariant Convolutional Networks.\" NeurIPS 2023.\\\n[2] Taco Cohen, and Max Welling. \"Group Equivariant Convolutional Networks.\" ICML 2016.\\\n[3] Hyunsu Kim, et al. \"Variational Partial Group Convolutions for Input-Aware Partial Equivariance of Rotations and Color-Shifts.\" ICML 2024.\\\n[3] Yulong  Yang, et al. \"Learning Color Equivariant Representations.\" ICLR 2024", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "* The paper is well motivated and provides an elegant solution.\n* SEqSI is provably equivariant to scale and invariant to shift. Furthermore, SEqSI achieves this with significantly less constraint than baseline line methods.\n* SEqSI introduces a pipeline and loss function that extends equivariance to thresholding tasks, enabling wider application to tasks such as object localization and binary segmentation.\n* SEqSI demonstrates strong performance compared to baseline methods on both classification and object localization task, both with and without data augmentation, and under affine and non-affine transformations.\n* SEqSI has computational efficiency significantly better than fully affine-equivariant baselines and on par with standard models without any equivariance and only scale-equivariance models.", "weaknesses": "While I believe this is overall a good paper that meets all ICLR standards for acceptance, I do think there are a few weaknesses that I would like the authors to address during rebuttal.\n1. The classification test in Table 3 is done on a relatively small and low resolution dataset (CIFAR-10). It would be more compelling if the authors could also present results on higher resolution (ie Stanford Cars, Oxford Pets, Caltech-101) and larger datasets (ImageNet). I don't think it is reasonable to run all experiments or the ImageNet dataset in the rebuttal time frame. If the authors can showcase results for non-augmented train set under only the four affine transformations for standard, SEqSI, and SEq for datasets such as Oxford Pets, that would convince me the scalability of the model.\n2. I think it would be cool to showcase binary segmentation results on a small toy dataset such as Caltech-101, if time permits.\n3. I am confused as to why Table 4 only includes non-equivariant baselines? While this demonstrates the benefits of equivariance, I don't think that was ever in question. Filling out the additional baselines would serve the paper much better.\n4. I think the authors should spend more time talking about the architectural differences between SEqSI and [Herbreteau, 2023], as they appear very similar on first glance, in terms of what they want to achieve and the ways they go about achieving them.", "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)