# Shift-Equivariant Complex-Valued Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2511.21250 · HTML: https://mlsanity.com/paper/2511.21250/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 53 references that are in the map.
- First posted on arXiv 26 Nov 2025.

## Builds on (most cited first, 19 of 19)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Making Convolutional Networks Shift-Invariant Again](https://mlsanity.com/paper/1904.11486/) (2019)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [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)
- [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)
- [Learnable Polyphase Sampling for Shift Invariant and Equivariant Convolutional Networks](https://mlsanity.com/paper/2210.08001/) (2022)
- [Making Vision Transformers Truly Shift-Equivariant](https://mlsanity.com/paper/2305.16316/) (2023)
- [Complex Fully Convolutional Neural Networks for MR Image Reconstruction](https://mlsanity.com/paper/1807.03343/) (2018)
- [Complex-Valued vs. Real-Valued Neural Networks for Classification Perspectives: An Example on Non-Circular Data](https://mlsanity.com/paper/2009.08340/) (2020)
- [PolSF: PolSAR image dataset on San Francisco](https://mlsanity.com/paper/1912.07259/) (2019)
- [Optimal approximation using complex-valued neural networks](https://mlsanity.com/paper/2303.16813/) (2023)
- [Learning Representations Using Complex-Valued Nets](https://mlsanity.com/paper/1511.06351/) (2015)

## Abstract (arXiv)

Convolutional neural networks have shown remarkable performance in recent years on various computer vision problems. However, the traditional convolutional neural network architecture lacks a critical property: shift equivariance and invariance, broken by downsampling and upsampling operations. Although data augmentation techniques can help the model learn the latter property empirically, a consistent and systematic way to achieve this goal is by designing downsampling and upsampling layers that theoretically guarantee these properties by construction. Adaptive Polyphase Sampling (APS) introduced the cornerstone for shift invariance, later extended to shift equivariance with Learnable Polyphase up/downsampling (LPS) applied to real-valued neural networks. In this paper, we extend the work on LPS to complex-valued neural networks both from a theoretical perspective and with a novel building block of a projection layer from mathbbC to ℝ before the Gumbel Softmax. We finally evaluate this extension on several computer vision problems, specifically for either the invariance property in classification tasks or the equivariance property in both reconstruction and semantic segmentation problems, using polarimetric Synthetic Aperture Radar images.