用几何建模解决联邦学习个性化难题，提升异构数据下性能与隐私。

Personalized federated learning (PFL) often fails under label skew and non-stationarity because a single global parameterization ignores client-specific geometry. We introduce VGM$^2$ (Variational Gaussian Mixture Manifold), a geometry-centric PFL framework that (i) learns client-specific parametric UMAP embeddings, (ii) models latent pairwise distances with mixture relation markers for same and different class pairs, and (iii) exchanges only variational, uncertainty-aware marker statistics. Each client maintains a Dirichlet-Normal-Inverse-Gamma (Dir-NIG) posterior over marker weights, means, and variances; the server aggregates via conjugate moment matching to form global priors that guide subsequent rounds. We prove that this aggregation minimizes the summed reverse Kullback-Leibler divergence from client posteriors within the conjugate family, yielding stability under heterogeneity. We further incorporate a calibration term for distance-to-similarity mapping and report communication and compute budgets. Across eight vision datasets with non-IID label shards, VGM$^2$ achieves competitive or superior test F1 scores compared to strong baselines while communicating only small geometry summaries. Privacy is strengthened through secure aggregation and optional differential privacy noise, and we provide a membership-inference stress test. Code and configurations will be released to ensure full reproducibility.