# From CNNs to Shift-Invariant Twin Models Based on Complex Wavelets

arXiv: https://arxiv.org/abs/2212.00394 · HTML: https://mlsanity.com/paper/2212.00394/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 50 references that are in the map.
- First posted on arXiv 1 Dec 2022.

## Builds on (most cited first, 30 of 32)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Benchmarking Neural Network Robustness to Common Corruptions and Perturbations](https://mlsanity.com/paper/1903.12261/) (2019)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [How transferable are features in deep neural networks?](https://mlsanity.com/paper/1411.1792/) (2014)
- [CvT: Introducing Convolutions to Vision Transformers](https://mlsanity.com/paper/2103.15808/) (2021)
- [Network In Network](https://mlsanity.com/paper/1312.4400/) (2013)
- [Making Convolutional Networks Shift-Invariant Again](https://mlsanity.com/paper/1904.11486/) (2019)
- [Escaping the Big Data Paradigm with Compact Transformers](https://mlsanity.com/paper/2104.05704/) (2021)
- [Incorporating Convolution Designs into Visual Transformers](https://mlsanity.com/paper/2103.11816/) (2021)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [Why do deep convolutional networks generalize so poorly to small image transformations?](https://mlsanity.com/paper/1805.12177/) (2018)
- [How Much Position Information Do Convolutional Neural Networks Encode?](https://mlsanity.com/paper/2001.08248/) (2020)
- [On Translation Invariance in CNNs: Convolutional Layers can Exploit Absolute Spatial Location](https://mlsanity.com/paper/2003.07064/) (2020)
- [Delving Deeper into Anti-aliasing in ConvNets](https://mlsanity.com/paper/2008.09604/) (2020)
- [Truly shift-invariant convolutional neural networks](https://mlsanity.com/paper/2011.14214/) (2020)
- [Deep Roto-Translation Scattering for Object Classification](https://mlsanity.com/paper/1412.8659/) (2014)
- [A Mathematical Theory of Deep Convolutional Neural Networks for Feature Extraction](https://mlsanity.com/paper/1512.06293/) (2015)
- [Graph Convolutional Neural Networks via Scattering](https://mlsanity.com/paper/1804.00099/) (2018)
- [Scaling the Scattering Transform: Deep Hybrid Networks](https://mlsanity.com/paper/1703.08961/) (2017)
- [Convolutional Neural Networks Are Not Invariant to Translation, but They Can Learn to Be](https://mlsanity.com/paper/2110.05861/) (2021)
- [Deep Network Classification by Scattering and Homotopy Dictionary Learning](https://mlsanity.com/paper/1910.03561/) (2019)
- [Parametric Scattering Networks](https://mlsanity.com/paper/2107.09539/) (2021)
- [Group Equivariant Subsampling](https://mlsanity.com/paper/2106.05886/) (2021)
- [A mathematical motivation for complex-valued convolutional networks](https://mlsanity.com/paper/1503.03438/) (2015)
- [Complex Fully Convolutional Neural Networks for MR Image Reconstruction](https://mlsanity.com/paper/1807.03343/) (2018)
- [A Learnable ScatterNet: Locally Invariant Convolutional Layers](https://mlsanity.com/paper/1903.03137/) (2019)
- [In What Ways Are Deep Neural Networks Invariant and How Should We Measure This?](https://mlsanity.com/paper/2210.03773/) (2022)

## Abstract (arXiv)

We propose a novel method to increase shift invariance and prediction accuracy in convolutional neural networks. Specifically, we replace the first-layer combination "real-valued convolutions + max pooling" (RMax) by "complex-valued convolutions + modulus" (CMod), which is stable to translations, or shifts. To justify our approach, we claim that CMod and RMax produce comparable outputs when the convolution kernel is band-pass and oriented (Gabor-like filter). In this context, CMod can therefore be considered as a stable alternative to RMax. To enforce this property, we constrain the convolution kernels to adopt such a Gabor-like structure. The corresponding architecture is called mathematical twin, because it employs a well-defined mathematical operator to mimic the behavior of the original, freely-trained model. Our approach achieves superior accuracy on ImageNet and CIFAR-10 classification tasks, compared to prior methods based on low-pass filtering. Arguably, our approach's emphasis on retaining high-frequency details contributes to a better balance between shift invariance and information preservation, resulting in improved performance. Furthermore, it has a lower computational cost and memory footprint than concurrent work, making it a promising solution for practical implementation.