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Convolution Goes Higher-Order: A Biologically Inspired Mechanism Empowers Image Classification
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Proposes higher-order (Volterra-like) convolutions that add learnable multiplicative polynomial terms to CNN kernels, showing gains on synthetic textures, MNIST/FashionMNIST/CIFAR, and ImageNet with ResNet-18.
Problem
Standard CNNs rely on linear convolutions followed by pointwise nonlinearities, which may fail to capture higher-order spatial correlations in images, especially in shallow or narrow networks.
Approach
Extend the convolution operator with learnable Volterra-like higher-order terms (2nd to 4th order products of patch pixels) summed into a single feature map before ReLU, so multiplicative interactions between neighboring pixels are modeled within one layer.
Key results
- On ImageNet, HoResNet-18 reaches 69.47% ± 0.09% vs ResNet-18 68.62% ± 0.08% over 5 runs.
- On CIFAR-10, HoCNN reaches 72.87% ± 0.54% vs CNN 69.76% ± 0.64% over 50 seeds, with 80,262 vs 82,706 parameters.
- On CUB-200-2011, HoResNet-18 reaches 72.31% vs ResNet-18 68.24% in a single run; on synthetic textures, accuracy rises from 59.14% (CNN) to 92.32% (up to 4th order).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace standard convolution with higher-order convolution up to 4th order | CNN · Synthetic texture test set | accuracy: 59.14 -> 92.32 | Helped |
| Use 3rd-order higher-order convolutions instead of standard convolutions | CNN · MNIST, FashionMNIST, CIFAR-10, CIFAR-100 | accuracy: 99.13 $\pm$ 0.09 -> 99.30 $\pm$ 0.07; 90.48 $\pm$ 0.32 -> 90.95 $\pm$ 0.35; 69.76 $\pm$ 0.64 -> 72.87 $\pm$ 0.54; 36.92 | Helped |
| Use higher-order residual blocks in the first stage | ResNet-18 · ImageNet | accuracy: 68.62\% $\pm$ 0.08\% -> 69.47\% $\pm$ 0.09\%; 0.85\% | Helped |
| Use higher-order residual blocks in the first stage | ResNet-18 · Imagenette | accuracy: 88.13 -> 89.30; +1.17\% | Helped |
| Use higher-order residual blocks in the first stage | ResNet-18 · CUB-200-2011 | accuracy: 68.24 -> 72.31; +4.07\% | Helped |
| Use higher-order convolution rather than a deep standard CNN at a comparable FLOPs budget | CIFAR-10 classifiers · CIFAR-10 | accuracy: 71.20 -> 72.06 | Helped |
| Limit higher-order expansion to 2nd order rather than 4th order (ablation) | HoCNN · Synthetic texture test set | accuracy: 92.32 -> 82.42 | Hurt |
| Limit higher-order expansion to 3rd order rather than 4th order (ablation) | HoCNN · Synthetic texture test set | accuracy: 92.32 -> 89.02 | Hurt |
| Extend benchmark higher-order kernels from 3rd order to 4th order (ablation) | HoCNN · MNIST, FashionMNIST, CIFAR-10, CIFAR-100 | accuracy | No clear effect |
| Move the higher-order block from the second to the third convolutional layer (ablation) | Higher-order CNN · CIFAR-10 | accuracy: 71.17 ± 0.51\% -> 70.29 ± 0.49\% | Hurt |
| Move the higher-order block from the second to the third convolutional layer (ablation) | Higher-order CNN · CIFAR-100 | accuracy: 37.27 ± 0.80\% -> 36.09 ± 0.81\% | Hurt |
| Replace standard baseline kernels with 5×5 or 7×7 kernels (ablation) | CNN · CIFAR-10 and CIFAR-100 | accuracy: CIFAR-10: 68.72 ± 0.63\% (5×5), 67.51 ± 0.61\% (7×7); CIFAR-100: 36.04 ± 0.76\% (5×5), 34.01 ± 0.72\% (7×7) | Hurt |
Surprising findings
- Higher-order CNNs are more sensitive to controlled higher-order statistical perturbations yet more robust to common CIFAR-10-C/CIFAR-100-C corruptions.
- Increasing kernel size in standard CNNs (5x5, 7x7) decreased accuracy rather than improving it.
Limitations
- Higher-order operations add substantial FLOPs and parameters, growing with order, limiting use beyond 4th order.
- Increased sensitivity to controlled higher-order statistical perturbations.
- Placing higher-order layers deeper degrades performance; no low-rank approximation used.
Experimental setup
Datasets: MNIST FashionMNIST CIFAR-10 CIFAR-100 ImageNet Imagenette CUB-200-2011 CIFAR-10-C CIFAR-100-C synthetic binary textures (Metex)
Benchmarks: MNIST FashionMNIST CIFAR-10 CIFAR-100 ImageNet Imagenette CUB-200-2011 CIFAR-10-C CIFAR-100-C synthetic texture glider classification
Metrics: test accuracy normalized accuracy representational dissimilarity matrix number of parameters FLOPs
Baselines: CNN ResNet-18 ResNet-34 ResNet-50 Deep CNN VOneNet
Builds on: Volterra expansion ResNet AdamW
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