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.
Gabriele D’Acunto, Enrico Grimaldi, Valeria Avino et al.· 0 citations
Modern sensing, communication, and learning systems generate heterogeneous network signals, with local data differing in dimension, modality, and geometric structure. Processing such data requires a mathematical framework capable of simultaneously modeling heterogeneous local signal spaces and the transformations relating them. Network sheaves provide such a framework by associating local vector spaces with network entities and linear restriction maps with their interactions. This is the first paper to develop a unified sheaf signal processing (SSP) framework on network sheaves, extending the fundamental operations of signal processing, namely spectral analysis, filtering, and sampling, to heterogeneous local spaces. Unlike graph and topological signal processing, where signals are modeled over a common vector space, SSP jointly models heterogeneous local signal spaces and the linear transformations relating neighboring spaces through restriction maps. We define the Sheaf Fourier Transform (SFT), whose frequencies quantify signal inconsistency induced by the network topology, the restriction maps, and the local geometry. Building on this representation, we develop polynomial sheaf filters and formulate sampling as the joint selection of network nodes and intra-node components. We derive perfect recovery conditions for bandlimited sheaf signals and propose a greedy sampling-set design algorithm. To incorporate application-dependent signal models, including different bases, dictionaries, and learned embeddings, we introduce representation sheaves and characterize the natural transformations that preserve spectral properties and guarantee interoperability across representations. Experiments on synthetic, motion-capture, and financial datasets validate the proposed framework and demonstrate consistent improvements over canonical graph signal processing baselines.
Gabriele D’Acunto, Leonardo Di Nino, P. Lorenzo et al.· 0 citations
Structured Connection Graph Learning (SCGL), a block-coordinate algorithm that combines closed-form updates, manifold projections, and spectral constraints, and converges to stationary points of the resulting nonconvex problem, is developed.
Leonardo Di Nino, Gabriele D’Acunto, Sergio Barbarossa et al.· 0 citations
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