Heterogeneous federated systems require agents to learn and exchange informative representations despite differences in data distributions, sensing modalities, model architectures, latent dimensionalities, and local learning objectives. To address this challenge, we propose Sheaf-based Federated Representation Learning (SFRL), a general framework that jointly optimizes local objectives with a manifold-constrained geometric alignment regularizer based on learnable sheaf restriction maps. Unlike most existing approaches, SFRL does not assume a shared global latent space. Instead, global consistency emerges from the alignment of neighboring latent representations through orthogonal transformations and isometric embeddings. This alignment is enforced by a quadratic gluing regularizer induced by the sheaf Laplacian, whose learnable restriction maps adapt the geometry to the observed data. The penalty is evaluated on a small set of shared pilot samples, ensuring scalability and communication efficiency. We develop a decentralized algorithm for solving SFRL, termed Sheaf-FRL, which alternates between gradient updates of the local models and closed-form Procrustes updates of the edge-wise restriction maps. We further establish convergence of Sheaf-FRL to first-order stationary points in both deterministic and stochastic settings. As an application, we consider a cooperative classification task in the context of semantic communication, under model and data heterogeneity. Our results show that Sheaf-FRL outperforms baseline approaches in terms of local and post-communication classification accuracy across different levels of local distribution shift and exhibits greater robustness to latent-space dimensionality compression.
Codebook-based blind face restoration (BFR) often suffers from ambiguous conditioning features and a fragile prediction mechanism under severe degradation. To address these challenges, we propose GeoMAR, a framework designed to unleash geometrically aligned features with masked autoregressive (MAR) refinement for robust face restoration. For feature conditioning, we introduce a dual-input extraction pipeline to extract component-based geometric descriptions with explicit, spatially faithful anchors. These textual priors are integrated with low-quality (LQ) features via an Aligned Geometric Priors Injector, which employs a KV-Q exchange strategy to generate geometrically aligned features. For prediction mechanism, we reformulate the one-step mapping into a multi-step MAR process. This coarse-to-fine generation progressively refines complex facial regions based on increasingly reliable context. Experiments on one synthetic and three real-world benchmarks demonstrate that GeoMAR achieves highly competitive perceptual quality and coherent visual structures compared with existing methods. The code is available at https://github.com/BRL-SYSU/GeoMAR.git.
Adversarial training is effective on balanced datasets, but its robustness degrades under longtailed class distributions, where tail classes suffer high robust error and unstable decision boundaries. We propose Manifold-Constrained Adversarial Training (MCAT), a unified framework that enforces the semantic validity of adversarial examples by penalizing deviations from class-conditional manifolds in feature space, while promoting balanced geometric separation across classes via an ETF-inspired regularization. We provide theoretical results that link geometric separation to lower bounds on adversarially robust margins, and show that manifold-constrained adversarial risk upperbounds robust risk on high-density semantic regions. Extensive experiments on standard longtailed benchmarks demonstrate consistent improvements in overall, balanced, and tail-class adversarial robustness.