# Publications

Publications in reversed chronological order. Generated by jekyll-scholar.

## 2022

- PreprintFrom CNNs to Shift-Invariant Twin Wavelet ModelsDec 2022
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" (RMax) by "complex-valued convolutions + modulus" (CMod), 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, CMod can be viewed as a stable alternative to RMax. 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.

- PreprintOn the Shift Invariance of Max Pooling Feature Maps in Convolutional Neural NetworksSep 2022
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.

## 2021

- WorkshopModélisation Parcimonieuse de CNNs Avec Des Paquets d’Ondelettes Dual-Tree
*In ORASIS,*Sep 2021We propose to improve the mathematical interpretability of convolutional neural networks (CNNs) for image classification. In this purpose, we replace the first layers of existing models such as AlexNet or ResNet by an operator containing the dual-tree wavelet packet transform, i.e., a redundant decomposition using complex and oriented waveforms. Our experiments show that these modified networks behave very similarly to the original models once trained. The goal is then to study this operator from a theoretical point of view and to identify potential optimizations. We want to analyze its main properties such as directional selectivity, stability with respect to small shifts and rotations, thus retaining discriminant information while decreasing intra-class variability. This work is a step toward a more complete description of CNNs using well-defined mathematical operators, characterized by a small number of arbitrary parameters, making them easier to interpret.