Authors: Emily J. King, Dustin G. Mixon, Michael Perlmutter, Lander Ver Hoef
Organizations: Department of Mathematics, Colorado State University, Fort Collins, CO, USA · Department of Mathematics and Translational Data Analytics Institute, The Ohio State University, Columbus, OH, USA · Department of Mathematics, Boise State University, Boise, ID, USA · Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, CO, USA
Many learning tasks require stability to small transformations while retaining sensitivity to larger ones. We introduce transversal pooling neural networks, which generalize spatial max pooling to affine group actions. We establish equivariance to a chosen subgroup and derive explicit stability bounds for individual pooled wavelet coefficients under affine perturbations of the input. Experimentally, we demonstrate the utility of our networks in low-data environments and for predicting tropical cyclone intensification.
Figures & tables
Figure 1: Four affine transformations of the leftmost digit ‘6.’ The middle three images result from transformations that are close to the identity. The rightmost image results from a 180 -degree rotation. This classification task (recognizing digits) is well suited for the transversal pooling architecture we introduce.
Figure 2: Left: A 6×4 grid of feature activations passes through a max pooling layer that converts it into a 3×2 grid of representative activations. Right: We apply max pooling to a continuous field of activations (specifically, f(x,y)=6x3−14y for x∈[0,2] and y∈[0,3] ) to reduce it to a 3×2 grid of representative activations. This variant of max pooling, which we denote by Ξ , is developed in the more general context of affine groups and transversals in Section 2.2 .
Figure 3: Illustration of a transversal pooling neural network. We process an input image f by first taking its cross-correlation with 12 different rotated versions of a single template before counter-rotating to produce 12 slices of a data cube. This implements the map Ψ with group G=C12⋉R2 (though due to the counter rotation, it is parametrized as Ψ(f)(A,Ab) rather than Ψ(f)(A,b) ). Next, we batch the slices into neighboring triples, re-rotate the slices in each batch to align with the middle slice, and then compute their pointwise maximum. This max pooling operation corresponds to the map Ξ with subgroup Λ=C4⋉R2 . Thanks to the counter-rotated parameterization of the intermediate data cube, rotating the input image by 30 degrees (resp. 90 degrees) has the effect of cyclically permuting the 12 intermediate slices (resp. four output slices).
Name
G
ΛD
ΛT
Λ≤G ?
Location
Rotations
SO(d)
ΛD≤SO(d)
{0}
✓
Example 3.7
Translations
Rd
{I}
cZd
✓
Example 3.9
Isotropic dilations
{sI:s>0}
aZI
{0}
✓
Example 3.10
Rototranslations
SO(2)⋉R2
C4
cZ2
✓
Example 3.11
Discrete rototranslations
C4⋉Z2
{I}
cZ2
✓
Example 3.12
Direct similitudes
(R>0×SO(d))⋉Rd
aZΛr , Λr≤SO(d)
cZd
✗
Example 3.16
Table 1: Ambient transformation groups G and retained index sets Λ={(A,b):A∈ΛD,b∈ΛT} in the examples of Sections 3.2 and 3.3 .
Figure 4: Left: A randomly generated ellipse image used for our experiments in Section 4.1 . Right: The maximum over λ∈Λ of the absolute difference between the pooled wavelet coefficients of an image and its rotated version, divided by the L2 norm of the unrotated image. The rotation angle θ is measured in degrees, and each colored curve corresponds to a different random trial.
Training Set Size
8000
4000
2000
800
400
80
40
16
CNN Acc.
98.4
97.5
98.6
97.6
48.9
51.1
51.1
49.7
Aug. CNN Acc.
98.5
98.9
98.1
96.6
48.9
51.1
59.5
51.2
TraPNet Acc.
98.7
98.5
98.9
98.4
97.2
75.5
65.7
51.7
Table 2: The accuracy of TraPNet models compared to baseline CNNs and augmented CNNs when trained on limited data for the orientation task described in Section 4.2 . The best results appear in bold. Since this is a binary task, 50% accuracy is as good as random guessing.
Accuracy
F1 score
Precision
Recall
Aug. CNN
0.929
0.264
0.278
0.252
TraPNet
0.926
0.328
0.305
0.354
Table 3: Test set results for predicting tropical cyclone rapid intensification. The best results appear in bold.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 5: Model architectures for the baseline CNN and the TraPNet used in Section 4.2 .
Figure 6: The architectures for the baseline CNN and the TraPNet used in Section 4.3 .
Convolutional Neural Networks (CNNs) are widely assumed to be translation-invariant, yet standard architectures exhibit a startling fragility: even a single-pixel shift can drastically degrade performance due to their reliance on spatially dependent fully connected layers. In this work, we resolve this vulnerability by proposing a lightweight 'Online Architecture' strategy. By strategically inserting Global Average Pooling (GAP) layers at various network depths, we effectively decouple feature recognition from spatial location. Using VGG-16 as a primary case study, we demonstrate that this architectural modification achieves a massive 98% reduction in trainable parameters (from 5.2M to just 82K) and a 90% reduction in total network size (138M to 14M). Despite this drastic pruning, our variants maintain competitive Top-1 accuracy on ImageNet (66.4%) while doubling translational robustness, reducing average relative loss from 0.09 to 0.05. Furthermore, our analysis identifies a fundamental limit to invariance: while GAP resolves macroscopic sensitivity, discrete pooling operations introduce a residual periodic aliasing that prevents perfect pixel-level stability. Finally, we extend these findings to Perceptual Image Quality Assessment (IQA) by integrating our invariant backbones into the LPIPS framework. The resulting metric significantly outperforms the retrained baseline in generalization across the KADID-10k dataset (Spearman 0.89 vs. 0.75) and achieves a near-perfect alignment with human psychophysical response curves on the RAID dataset (Spearman 0.95). These results confirm that enforcing architectural invariance is a far more efficient and biologically plausible path to robustness than traditional data augmentation. Data and code are publicly available. The data and code are publicly available to facilitate validation and further research.
Nuria Alabau-Bosque, Jorge Vila-Tomas, Paula Dauden-Oliver +2
Image Processing Lab, Universitat de València, Carrer del Catedr´atic Jos´e Beltr´an Martinez, Paterna, 46980, Spain.
The universal approximation theorem establishes that neural networks can approximate any continuous function on a compact set. Later works in approximation theory provide quantitative approximation rates for ReLU networks on the class of α-Hölder functions f:[0,1]N→R. The goal of this paper is to provide similar quantitative approximation results in the context of group equivariant learning, where the learned α-Hölder function is known to obey certain group symmetries. While there has been much interest in the literature in understanding the universal approximation properties of equivariant models, very few quantitative approximation results are known for equivariant models. In this paper, we bridge this gap by deriving quantitative approximation rates for several prominent group-equivariant and invariant architectures. The architectures that we consider include: the permutation-invariant Deep Sets architecture; the permutation-equivariant Sumformer and Transformer architectures; joint invariance to permutations and rigid motions using invariant networks based on frame averaging; and general bi-Lipschitz invariant models. Overall, we show that equally-sized ReLU MLPs and equivariant architectures are equally expressive over equivariant functions. Thus, hard-coding equivariance does not result in a loss of expressivity or approximation power in these models.
Jonathan W. Siegel, Snir Hordan, Hannah Lawrence +2
Department of Mathematics, Texas A&M University, College Station, TX, 77840, USA · Faculty of Mathematics, Technion- Israel Institute of Technology, Haifa, 3200003, Israel · Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, 02139 USA
We introduce Lie groupoid equivariant neural networks as a specialization of recently proposed topological category-equivariant neural networks to the differentiable setting. Lie groupoid equivariant neural networks are composed from Lie groupoid lifting convolutions and Lie groupoid convolution layers, and we show how for suitable Lie groupoids they are equivalent to certain Lie algebroid-equivariant neural networks. We additionally describe groupoid invariant global pooling as a generalization of group invariant global pooling. Furthermore, we show that each of the aforementioned layers is a special case of recently introduced admissible category-equivariant layers by demonstrating that they define continuous natural transformations between continuous feature functors.
Michael Astwood
Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2M6 Canada