🤖 AI Summary
This study investigates under what conditions complex-valued neural networks (CVNNs) genuinely outperform their real-valued counterparts, clarifying the relationship between input complex structure and learning advantages. Employing a representation-first evaluation framework, the authors systematically compare CVNNs against parameter- and FLOP-matched real-valued baselines across radio frequency, quantum wavefunction, and EEG tasks, complemented by analyses involving polar/Cartesian coordinate transformations, CReLU activation, gradient dynamics, and hyperparameter ablation. The findings reveal that CVNNs’ superiority stems from specific representational structures, symmetries, and optimization properties—not inherent architectural dominance—and is most pronounced in PSK modulation tasks, whereas magnitude-based real models excel in QAM. After independent hyperparameter tuning, CVNNs achieve only a 2.46 percentage point gain on RadioML, substantially lower than the previously reported 22.94, indicating that performance gaps are largely attributable to hyperparameter sensitivity.
📝 Abstract
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 $|ψ|$ 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 $\times$ 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.