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

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 ...

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