Averaging Models Is Not Averaging Spectra: Steering Federated Representations

Federated learning averages model parameters, whereas representation diversity belongs to the features produced by the aggregated model. We study the gap between these two objects through class-conditional von Neumann entropy (VNE). Exact linear constructions reveal a two-sided phenomenon: parameter averaging can either increase or decrease VNE relative to the local encoders. This spectral ambiguity leads to a second question—how far can the aggregated representation depart from centralized training? Under explicit regularity and heterogeneity assumptions, we bound the one-round diagnostic deviation from a same-start centralized reference, including rank-deficient empirical covariances. We then turn from diagnosis to control: interpolation toward a shared class-conditional density matrix contracts a weighted spectral disagreement, motivating the class-conditional FedVNA regularizer. On CIFAR-10 with ResNet-18 and Dirichlet $\alpha =0.1$, FedVNA raises final VNE from FedAvg's $3.0655$ to $4.0782$, while their best accuracies remain close ($0.7464$ and $0.7392$). Together, the results show that federated spectra can be diagnosed and deliberately shaped, while spectral diversity and predictive performance remain distinct outcomes.