From CNNs to Shift-Invariant Twin Wavelet Models

We propose a novel antialiasing method to increase shift invariance in convolutional neural networks (CNNs). More precisely, we replace the conventional combination "real-valued convolutions + max pooling" ($\mathbb R$Max) by "complex-valued convolutions + modulus" ($\mathbb C$Mod), which produce stable feature representations for band-pass filters with well-defined orientations. In a recent work, we proved that, for such filters, the two operators yield similar outputs. Therefore, $\mathbb C$Mod can be viewed as a stable alternative to $\mathbb R$Max. To separate band-pass filters from other freely-trained kernels, in this paper, we designed a "twin" architecture based on the dual-tree complex wavelet packet transform, which generates similar outputs as standard CNNs with fewer trainable parameters. In addition to improving stability to small shifts, our experiments on AlexNet and ResNet showed increased prediction accuracy on natural image datasets such as ImageNet and CIFAR10. Furthermore, our approach outperformed recent antialiasing methods based on low-pass filtering by preserving high-frequency information, while reducing memory usage.

On the Shift Invariance of Max Pooling Feature Maps in Convolutional Neural Networks

In this paper, we aim to improve the mathematical interpretability of convolutional neural networks for image classification. When trained on natural image datasets, such networks tend to learn parameters in the first layer that closely resemble oriented Gabor filters. By leveraging the properties of discrete Gabor-like convolutions, we prove that, under specific conditions, feature maps computed by the subsequent max pooling operator tend to approximate the modulus of complex Gabor-like coefficients, and as such, are stable with respect to certain input shifts. We then compute a probabilistic measure of shift invariance for these layers. More precisely, we show that some filters, depending on their frequency and orientation, are more likely than others to produce stable image representations. We experimentally validate our theory by considering a deterministic feature extractor based on the dual-tree wavelet packet transform, a particular case of discrete Gabor-like decomposition. We demonstrate a strong correlation between shift invariance on the one hand and similarity with complex modulus on the other hand.