FedSPDnet: Geometry-Aware Federated Deep Learning with SPDnet

Abstract: We introduce two federated learning frameworks for the classical SPDnet model operating on symmetric positive definite (SPD) matrices with Stiefel-constrained parameters. Unlike standard Euclidean averaging, which violates orthogonality, our approach preserves geometric structure through two efficient aggregation strategies: ProjAvg, projecting arithmetic means onto the Stiefel manifold, and RLAvg, approximating tangent-space averaging via retractions and liftings. Both methods are computationally efficient, independent of the optimizer, and enable scalable federated learning for signal processing applications whose features are SPD matrices. Simulations on EEG motor imagery benchmarks show that FedSPDnet outperforms federated EEGnet in F1 score and robustness to federation and partial participation, while using fewer parameters per communication round.

• The paper introduces geometry-preserving aggregation protocols (ProjAvg and RLAvg) that maintain the orthogonality of SPDnet's parameters during federated learning.
• It demonstrates that FedSPDnet outperforms conventional FedAvg on EEG benchmarks, achieving higher test F1 scores and improved robustness.
• The work provides a practical framework for deploying Riemannian neural networks in federated settings, balancing computational efficiency with theoretical rigor.

Geometry-Aware Federated Learning for SPDnet: The FedSPDnet Architecture

Motivation and Context

Federated learning (FL) typically aggregates client-trained models through parameter averaging in Euclidean space, which is not directly compatible with models whose parameters obey non-Euclidean constraints. For architectures such as SPDnet, wherein the core parameters are orthogonal matrices on the Stiefel manifold—a Riemannian manifold of rectangular orthonormal matrices—naive Euclidean averaging destroys manifold structure, violating orthogonality and degrading learned geometry. As interest grows in deploying FL across domains like brain-computer interface (BCI) applications that heavily utilize structured data (e.g., symmetric positive definite (SPD) matrices derived from EEG), the need for geometry-preserving, scalable FL frameworks has become acute.

Federated Learning on Manifolds: Problem and Prior Work

Conventional FL approaches, exemplified by FedAvg, are ill-suited for models like SPDnet, whose parameters live on non-Euclidean spaces. Several recent works extend FL to Riemannian manifolds [li2022federated, zhang2024nonconvex, huang2024riemannian], but these are broadly formulated and do not provide efficient, architecture-tailored solutions that account for the complex geometric structure of SPDnet's Stiefel parameters. Moreover, computing the Riemannian mean (Karcher mean) on the Stiefel is computationally burdensome and impractical for scaling.

The FedSPDnet Design: Geometry-Aware Aggregation

Orthogonality-Preserving Aggregation

FedSPDnet introduces two aggregation protocols for federated learning with SPDnet:

1. ProjAvg (Projection Averaging):
   • Client updates for each BiMap matrix $\MAT{W}_{\ell}$ are simply averaged in $\mathbb{R}^{p \times k}$, followed by a projection back to the Stiefel manifold using the orthogonal factor from the polar decomposition. This results in a computational complexity linear in the number of clients and dimensions, with no need to store previous iterates or compute Riemannian exponential/logarithm mappings.
   • The method does not require specialized optimization methods on clients; any optimizer (e.g., Adam, SGD) can be used.
2. RLAvg (Retraction-Lifting Averaging):
   • This scheme is an approximation to Riemannian averaging: each client’s updated matrix is mapped to the tangent space at the last global model via a “lifting” (using the tangent projection), then averaged in that tangent space, and finally “retracted” back onto the Stiefel.
   • Like ProjAvg, RLAvg is agnostic to the local optimizer and maintains orthogonality, but requires retaining and referencing the previous global iterate at aggregation.

For the Euclidean softmax parameters in SPDnet, traditional FedAvg suffices.

Theoretical Properties

Neither ProjAvg nor RLAvg directly solve Riemannian barycenter problems; rather, they provide tractable, efficient surrogates that respect the geometric constraints inherent in SPDnet. Both methods scale efficiently with the number of clients, do not depend on specialized client optimizers, and—critically—prevent "leakage" of information from outside the manifold (which is crucial for convergence and stability with non-Euclidean architectures).

Experimental Setup and Results

FedSPDnet is empirically validated on EEG-based motor imagery classification benchmarks, which are representative of sensitive applications with distributed cohorts, non-i.i.d. data, and privacy constraints. The experiments compare FedSPDnet (SPDnet with ProjAvg or RLAvg) to the Euclidean FL baseline (FedEEGnet, i.e., EEGnet with FedAvg) over the Weibo2014 and PhysionetMI datasets.

Key settings include:

• Signal processing to obtain SPD matrices per trial (for SPDnet) and raw trials (for EEGnet).
• Cross-entropy loss minimized via Adam in all settings.
• Full (100%) and partial ($\lfloor N/2 \rfloor$) client participation scenarios to test robustness to federation and missing data.

Strong numerical results include:

• FedSPDnet with either aggregation method consistently outperforms FedEEGnet in test F1, robustness to partial participation, and convergence speed.
• On Weibo2014, FedSPDnet achieves $43.3 \pm 1.0\%$ test F1 (full participation), exceeding the $39.9 \pm 2.4\%$ of FedEEGnet, and shows a smaller drop from the centralized model ($51.7 \pm 0.8\%$).
• On PhysionetMI, although centralized EEGnet is superior, FedSPDnet narrows the gap in the federated setting, retaining more performance and demonstrating less sensitivity to heterogeneity.
• ProjAvg and RLAvg are numerically equivalent, as evidenced by overlapping F1 trajectories over all rounds and data splits, with ProjAvg slightly preferred for implementation simplicity and lower memory usage.

Implications and Future Directions

Practical Use

The principal practical implication is that geometry-aware aggregation in FL can match or exceed conventional Euclidean methods even when the target architecture requires complex geometric constraints. FedSPDnet enables federated deployment of Riemannian neural networks like SPDnet with minimal changes to existing FL infrastructure, broadening the scope for FL in domains where manifold-valued data is native (e.g., neuroimaging, radar, remote sensing).

Theoretical Impact

These results underline the importance of respecting parameter geometry in distributed optimization, pointing to a gap between general-purpose manifold FL theory and architecture-specific, efficient implementations. The presented aggregation schemes advance the empirical frontier for scalable learning with orthogonal and SPD constraints.

Open Questions and Research Avenues

• Global Convergence and Generalization: While empirical convergence is robust, analytical convergence and generalization guarantees under non-convex, non-Euclidean FL remain largely open.
• Heterogeneous Manifolds and Privacy: Future work may address statistical and system heterogeneity, as well as differential privacy, when clients’ data or model structures are themselves manifold-valued or hybrid.
• Continuous and Adaptive Participation: Exploring dynamic participation rates, client sampling strategies, and asynchronous communication with geometry-aware aggregation constitutes a future direction, as real FL deployments rarely adhere to synchronized rounds.

Conclusion

FedSPDnet establishes a practical and theoretically motivated framework for federated deep learning with SPDnet by introducing two efficient, optimizer-independent aggregation schemes that preserve Stiefel manifold constraints. The work demonstrates compelling numerical advantages over standard Euclidean FL baselines across challenging non-i.i.d. neuroimaging datasets and provides a blueprint for extending FL to Riemannian neural architectures. This paves the way for principled, federated deployment of geometry-aware deep learning models in privacy-sensitive, multi-institutional applications.