摘要：异构联邦系统需要智能体在数据分布、传感模态、模型架构、潜在维度以及本地学习目标存在差异的情况下，学习并交换具有信息量的表征。为解决这一挑战，我们提出基于层的联邦表征学习（Sheaf-based Federated Representation Learning，SFRL），这是一个通用框架，可联合优化本地目标，以及一种基于可学习层限制映射、受流形约束的几何对齐正则项。不同于大多数现有方法，SFRL 不假设存在共享的全局潜在空间。相反，全局一致性通过正交变换和等距嵌入对相邻潜在表征进行对齐而自然形成。这种对齐由层拉普拉斯算子诱导的二次拼接正则项强制实现，其可学习的限制映射能够使几何结构适应观测数据。该惩罚项在一小组共享的先导样本上计算，从而确保可扩展性和通信效率。我们开发了一种用于求解 SFRL 的去中心化算法，称为 Sheaf-FRL；该算法在本地模型的梯度更新与边级限制映射的闭式 Procrustes 更新之间交替进行。我们进一步证明，Sheaf-FRL 在确定性和随机设置下均收敛至一阶驻点。作为应用，我们在模型和数据均存在异构性的条件下，考虑了语义通信场景中的协作分类任务。结果表明，在不同程度的本地分布偏移下，Sheaf-FRL 在本地分类准确率和通信后分类准确率方面均优于基线方法，并且对潜在空间维度压缩表现出更强的鲁棒性。

Abstract: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.