cs.CVMar 17, 2026

Accurate Shift Invariant Convolutional Neural Networks Using Gaussian-Hermite Moments

Authors: Jaspreet Singh, Petra Bosilj, Grzegorz Cielniak

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Abstract

The convolutional neural networks (CNNs) are not inherently shift invariant or equivariant. The downsampling operation, used in CNNs, is one of the key reasons which breaks the shift invariant property of a CNN. Conversely, downsampling operation is important to improve computational efficiency and increase the area of the receptive field for more contextual information. In this work, we propose Gaussian-Hermite Sampling (GHS), a novel downsampling strategy designed to achieve accurate shift invariance. GHS leverages Gaussian-Hermite polynomials to perform shift-consistent sampling, enabling CNN layers to maintain invariance to arbitrary spatial shifts prior to training. When integrated into standard CNN architectures, the proposed method embeds shift invariance directly at the layer level without requiring architectural modifications or additional training procedures. We evaluate the proposed approach on CIFAR-10, CIFAR-100, and MNIST-rot datasets. Experimental results demonstrate that GHS significantly improves shift consistency, achieving 100% classification consistency under spatial shifts, while also improving classification accuracy compared to baseline CNN models.

Figures & tables

Figure 1: Illustration of invariance (left) and equivariance (right). In invariance, the operator output remains unchanged under the transformation. In equivariance, applying the operator after a transformation yields the same result as transforming the operator output.
Figure 2: Gaussian-Hermite polynomials of order 0 to 4 with Gaussian weighting (σ=0.75)(\sigma=0.75) .
Figure 3: GHS process for an image. (a) Unshifted ( F\mathbf{F} ) and shifted ( F′\mathbf{F}^{\prime} ) images with their maximum locations. (b) Construction of the GHM Amax⁡∗(F)\mathbf{A}^{*}_{\max}(\mathbf{F}) using Q\mathbf{Q} and QT\mathbf{Q}^{T} . (c) Downsampled image F^∗\widehat{\mathbf{F}}^{*} obtained using (Eq. ( 22 )).
Figure 4: Comparison of downsampling methods. (a) Original image, (b-e) unshifted downsampled results, (f) shifted image and (g-j) shifted downsampled results.
MethodResNet-18ResNet-20ResNet-50ResNet-56
Baseline90.88 ±0.16\pm{0.16}90.83 ±0.14\pm{0.14}88.96 ±0.14\pm{0.14}91.89 ±0.31\pm{0.31}
LPF98.10 ±0.12\pm{0.12}96.53 ±0.13\pm{0.13}97.38 ±0.20\pm{0.20}96.90 ±0.19\pm{0.19}
APS100.00 ±0.00\pm{0.00}100.00 ±0.00\pm{0.00}100.00 ±0.00\pm{0.00}100.00 ±0.00\pm{0.00}
GHS100.00 ±0.00\pm{0.00}100.00 ±0.00\pm{0.00}100.00 ±0.00\pm{0.00}100.00 ±0.00\pm{0.00}
Table 1: Classification consistency on CIFAR-10 test set using ResNet models with baseline, LPF, APS, and GHS downsampling methods. Best results are highlighted in bold.
MethodResNet-18ResNet-20ResNet-50ResNet-56
Baseline91.96 ±0.31\pm{0.31}89.76 ±0.26\pm{0.26}90.05 ±0.18\pm{0.18}91.40 ±0.20\pm{0.20}
LPF94.28 ±0.29\pm{0.29}91.56 ±0.30\pm{0.30}94.12 ±0.32\pm{0.32}92.98 ±0.23\pm{0.23}
APS94.48 ±0.17\pm{0.17}91.75 ±0.23\pm{0.23}94.07 ±0.25\pm{0.25}92.93 ±0.29\pm{0.29}
GHS95.54 ±0.13\pm{0.13}91.72 ±0.18\pm{0.18}94.40 ±0.17\pm{0.17}93.13 ±0.20\pm{0.20}
Table 2: Classification accuracy on CIFAR-10 test set using ResNet models with baseline, LPF, APS, and GHS downsampling methods. Best results are highlighted in bold.
MethodConsistencyAccuracy
Baseline84.3575.59
LPF90.5476.85
APS100.0077.12
GHS100.0077.92
Table 3: Consistency and accuracy on the CIFAR-100 test set using the ResNet-18 model.
MethodConsistencyAccuracy
UnflipFlipUnflipFlip
DA97.8484.9494.2244.97
LPS98.1989.2194.2846.21
APS100.00100.0094.4847.55
GHS100.00100.0095.0249.68
Table 4: Consistency and accuracy using ResNet-18 on unflipped and flipped CIFAR-10.
Figure 5: Consistency (left) and accuracy (right) of ResNet-18 with different downsampling methods on CIFAR-10 under random patch erasures.
Table 10
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