# Co-domain Symmetry for Complex-Valued Deep Learning

arXiv: https://arxiv.org/abs/2112.01525 · HTML: https://mlsanity.com/paper/2112.01525/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 59 references that are in the map.
- First posted on arXiv 2 Dec 2021.

## Builds on (most cited first, 29 of 29)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Colorful Image Colorization](https://mlsanity.com/paper/1603.08511/) (2016)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Long-tailed Recognition by Routing Diverse Distribution-Aware Experts](https://mlsanity.com/paper/2010.01809/) (2020)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Deep Scale-spaces: Equivariance Over Scale](https://mlsanity.com/paper/1905.11697/) (2019)
- [Rotation equivariant vector field networks](https://mlsanity.com/paper/1612.09346/) (2016)
- [Compact Bilinear Pooling](https://mlsanity.com/paper/1511.06062/) (2015)
- [Understanding Deep Convolutional Networks](https://mlsanity.com/paper/1601.04920/) (2016)
- [Superposition of many models into one](https://mlsanity.com/paper/1902.05522/) (2019)
- [Associative Long Short-Term Memory](https://mlsanity.com/paper/1602.03032/) (2016)
- [Better than Real: Complex-valued Neural Nets for MRI Fingerprinting](https://mlsanity.com/paper/1707.00070/) (2017)
- [Neuronal Synchrony in Complex-Valued Deep Networks](https://mlsanity.com/paper/1312.6115/) (2013)
- [Complex-valued Neural Networks with Non-parametric Activation Functions](https://mlsanity.com/paper/1802.08026/) (2018)
- [Complex Fully Convolutional Neural Networks for MR Image Reconstruction](https://mlsanity.com/paper/1807.03343/) (2018)
- [Tied Block Convolution: Leaner and Better CNNs with Shared Thinner Filters](https://mlsanity.com/paper/2009.12021/) (2020)
- [SurReal: Complex-Valued Learning as Principled Transformations on a Scaling and Rotation Manifold](https://mlsanity.com/paper/1910.11334/) (2019)

## Built on by (most cited first, 3 of 3)

- [Multi-Spectral Image Classification with Ultra-Lean Complex-Valued Models](https://mlsanity.com/paper/2211.11797/) (2022)
- [Fully complex-valued deep learning model for visual perception](https://mlsanity.com/paper/2212.07146/) (2022)
- [Normalize Filters! Classical Wisdom for Deep Vision](https://mlsanity.com/paper/2506.04401/) (2025)

## Abstract (arXiv)

We study complex-valued scaling as a type of symmetry natural and unique to complex-valued measurements and representations. Deep Complex Networks (DCN) extends real-valued algebra to the complex domain without addressing complex-valued scaling. SurReal takes a restrictive manifold view of complex numbers, adopting a distance metric to achieve complex-scaling invariance while losing rich complex-valued information. We analyze complex-valued scaling as a co-domain transformation and design novel equivariant and invariant neural network layer functions for this special transformation. We also propose novel complex-valued representations of RGB images, where complex-valued scaling indicates hue shift or correlated changes across color channels. Benchmarked on MSTAR, CIFAR10, CIFAR100, and SVHN, our co-domain symmetric (CDS) classifiers deliver higher accuracy, better generalization, robustness to co-domain transformations, and lower model bias and variance than DCN and SurReal with far fewer parameters.