# When do complex-valued neural networks help? A study of representation, geometry, and optimization

arXiv: https://arxiv.org/abs/2605.27673 · HTML: https://mlsanity.com/paper/2605.27673/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 28 references that are in the map.
- First posted on arXiv 26 May 2026.

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

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Deep Reinforcement Learning that Matters](https://mlsanity.com/paper/1709.06560/) (2017)
- [EEGNet: A Compact Convolutional Network for EEG-based Brain-Computer Interfaces](https://mlsanity.com/paper/1611.08024/) (2016)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Are GANs Created Equal? A Large-Scale Study](https://mlsanity.com/paper/1711.10337/) (2017)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [Accounting for Variance in Machine Learning Benchmarks](https://mlsanity.com/paper/2103.03098/) (2021)
- [Over the Air Deep Learning Based Radio Signal Classification](https://mlsanity.com/paper/1712.04578/) (2017)
- [Convolutional Radio Modulation Recognition Networks](https://mlsanity.com/paper/1602.04105/) (2016)
- [On the State of the Art of Evaluation in Neural Language Models](https://mlsanity.com/paper/1707.05589/) (2017)
- [Full-Capacity Unitary Recurrent Neural Networks](https://mlsanity.com/paper/1611.00035/) (2016)
- [Understanding and Improving Convolutional Neural Networks via Concatenated Rectified Linear Units](https://mlsanity.com/paper/1603.05201/) (2016)
- [Better than Real: Complex-valued Neural Nets for MRI Fingerprinting](https://mlsanity.com/paper/1707.00070/) (2017)
- [On Complex Valued Convolutional Neural Networks](https://mlsanity.com/paper/1602.09046/) (2016)
- [Modulation Pattern Detection Using Complex Convolutions in Deep Learning](https://mlsanity.com/paper/2010.15556/) (2020)

## Built on by (most cited first, 1 of 1)

- [IQ-JEPA: A Joint-Embedding Predictive Architecture with a Hermitian Vision Transformer for Sound Speed and Attenuation Estimation from Ultrasound IQ Data](https://mlsanity.com/paper/2607.22351/) (2026)

## Abstract (arXiv)

Complex-valued Neural Networks (CVNNs) are often motivated by domains where information is naturally encoded in magnitude and phase. Yet complex-valued inputs alone do not determine when complex arithmetic improves learning: the label signal may lie in amplitude, phase, their coupling, or a symmetry that real-valued models can also represent under suitable coordinates. We study this through a representation-first evaluation of CVNNs against Cartesian real, polar, phase-only, magnitude-only, parameter-matched real, and FLOP-matched real baselines. Across synthetic RF tasks, complex representations are useful but not universally superior. PSK-only tasks favor phase-aware and complex-valued models, QAM-only tasks favor magnitude-based models, mixed PSK+QAM gives only a small complex-valued advantage, and unseen carrier-phase rotations break coordinate-dependent models without augmentation. Similar patterns appear beyond RF: in quantum-wavefunction prediction, momentum is invisible to |psi| but recoverable from phase, while EEG analytic-signal experiments show that phase locking, amplitude bursts, and phase-amplitude coupling each favor different coordinate views. We also identify a benchmarking artifact on RadioML 2018.01A. Under matched-shared-trial selection, a CReLU complex model exceeds the best real baseline by 22.94 PP; under independent per-family tuning on the same data and 16-trial search space, the gap collapses to 2.46 PP. Gradient analysis traces the inflated gap to high-learning-rate first-step instability in real baselines, while complex parameter coupling distributes the loss signal more robustly. A learning-rate × activation factorial confirms the failure is primarily hyperparameter-driven. Overall, CVNNs are best viewed as structured inductive biases whose gains depend on representation, symmetry, and optimization, not as universally superior architectures.