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 .
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