A Crossover, Not a Constant: When Complex-Valued Networks Actually Help?

Complex-valued neural networks (CVNNs) are widely motivated in domains where the information is carried in magnitude and phase. The paper asks when that motivation is borne out and finds that the answer is a narrow regime and not a general advantage. On a geometry-controlled seven-class RadioML 2018.01A task, covering phase-dominant, amplitude-dominant, and joint modulations across nine SNR levels, a complex model beats the best real baseline by $7.94$ pp against the Cartesian-only baseline set conventionally used in this literature, and by $0.71$ pp once a polar real baseline is included, then loses to it by $0.58$ pp under independent per-family tuning of the same search space, and by $3.29$ pp under a stable complex activation. What is left is not a fixed advantage but a crossover: complex arithmetic is advantageous up to $12$ pp in the marginal-SNR regime where phase estimates are noisy and is overtaken by an explicit polar representation as soon as the signal is clean. For all three configurations we test, the crossover is between $2$ and $6$ dB, regardless of the class composition and the sequence length. Three separate confounds inflate the apparent advantage outside this regime: (i) an impoverished baseline set, under which the capacity-matched Cartesian baselines become seed-unstable at the selected learning rate while polar and magnitude baselines do not; (ii) the selection rule, since matched-shared-trial selection measures robustness to a shared trial rather than peak performance; and (iii) training budget, since a $9.3$ pp complex advantage on phase-shift-keying data falls to $0.4$ pp at convergence. A fourth axis is not a confound but a design choice with confound-sized effects: complex activation moves accuracy by up to $61$ pp on amplitude-structured EEG tasks, and the sign of the effect is task-dependent. Especially, $|z|$-gated activations recover amplitude structure but degrade phase-dominant tasks across RF, quantum, and EEG data, so magnitude-gating is a diagnostic to report rather than a prescription to follow. Finally, we reinterpret the complex layer architecture as a $U(1)$-equivariant subspace of a stacked-real two-channel layer architecture, not a gain but a reduction in expressiveness in line with symmetry, and test this interpretation by constructing a constrained real network and finding that it matches the complex network to a tolerance of $-0.9$ pp while outperforming a matched unconstrained real network by $2.4$ to $11.1$ pp. In attributing results to complex-valued arithmetic, it is the symmetry constraint that matters, not the data type.