FedPHM: Communication-Efficient Personalization through Parameterized Hypercomplex Multiplication

Personalized federated learning requires transferable shared information and client-specific adaptation under a limited communication budget. We propose FedPHM, a structured selective transmission framework based on Parameterized Hypercomplex Multiplication (PHM). Specifically, we decompose fully-connected layers into shared basis matrices and client-specific private interaction matrices via Kronecker factorization: shared bases and other shared network parameters are transmitted for aggregation, while private interactions persist locally. Per-step spectral projection bounds the interaction matrices and their influence on shared gradients. Theoretically, we establish a finite-time joint-stationarity bound for the PHM subsystem under partial client participation. With a horizon-dependent common step size, the residual is bounded by $O(T^{-1/2})+O({\nu }_A)$, where ${\nu }_A$ bounds the private-gradient noise variance. An exact quadratic-risk decomposition further separates the approximation cost of shared bases from the benefit of private interactions. Experiments on CIFAR-10, CIFAR-100, and FMNIST demonstrate that FedPHM achieves excellent personalized accuracy with less communication, and is robust under high heterogeneity and low participation scenarios.