FedSPDnet: Geometry-Aware Federated Deep Learning with SPDnet

Federated learning is a method that enables collaborative model training without centralizing raw data, typically achieved by iteratively averaging parameters on a central server. This approach allows multiple clients to participate in model training while preserving data privacy. With the increasing importance of data privacy, federated learning has gained significant attention in both academia and industry. However, traditional federated learning methods are mainly limited to Euclidean parameter spaces and cannot extend to non-Euclidean geometries, limiting their effectiveness in certain applications. Meanwhile, the rise of Riemannian optimization has provided new solutions for learning problems with geometric constraints, especially when dealing with symmetric positive definite (SPD) covariance matrices. SPDnet is an architecture that integrates bilinear mappings and nonlinear feature rectification on the Stiefel manifold, proving effective in applications such as micro-Doppler radar and electroencephalography.

The limitations of traditional federated learning methods in non-Euclidean geometries present a significant challenge. Specifically, when dealing with symmetric positive definite (SPD) matrices, standard Euclidean averaging disrupts orthogonality, leading to the loss of geometric structures. This is a major bottleneck for signal processing applications that require preserving geometric information. Additionally, existing methods face computational complexity in calculating the Riemannian mean, making them difficult to apply in large-scale federated learning. Therefore, effectively preserving geometric structures in federated learning, especially when handling SPD matrices, is an important and challenging problem.

FedSPDnet's core innovations lie in its two geometry-aware aggregation strategies: ProjAvg and RLAvg.

• �� ProjAvg: Projects arithmetic means onto the Stiefel manifold via polar decomposition, preserving orthogonality. This method is computationally efficient and suitable for large-scale federated learning.

• �� RLAvg: Approximates tangent-space averaging using retractions and liftings, avoiding the complexity of directly calculating the Riemannian mean. This method is independent of specific optimizers, enhancing its flexibility across different application scenarios.

These innovations provide new theoretical insights and demonstrate practical effectiveness, especially when dealing with data with geometric constraints.

FedSPDnet's methodology involves several key steps:

• �� Data Preparation: Experiments are conducted using the Weibo2014 and PhysionetMI datasets, containing 7 and 4 motor imagery classes, respectively.

• �� Model Architecture: The SPDnet network is employed, processing SPD matrices through bilinear mappings and nonlinear feature rectification.

• �� Aggregation Strategies:
• ProjAvg: Computes Euclidean averages of local weights and maps them back onto the Stiefel manifold via polar decomposition.
• RLAvg: Uses retractions and liftings to approximate tangent-space averaging, avoiding direct computation of the Riemannian mean.

• �� Experimental Setup: In each communication round, a subset of clients is selected for local training, and updated parameters are aggregated on the server.

The experimental design includes using two motor imagery datasets: Weibo2014 and PhysionetMI. Each dataset's signals are band-pass filtered, and the sample covariance matrix is used as the input feature. The baseline model employed is EEGnet, federated using the standard FedAvg strategy. Key hyperparameters include learning rate, batch size, and the number of local training epochs. Ablation studies are conducted to evaluate the effectiveness of different aggregation strategies. By comparing the performance of ProjAvg and RLAvg on different datasets, the effectiveness of FedSPDnet is validated.

Experimental results show that FedSPDnet performs excellently on multiple datasets, particularly in terms of F1 score and robustness. Specifically, ProjAvg and RLAvg exhibit similar convergence on Weibo2014 and PhysionetMI datasets, with validation F1 curves overlapping throughout all communication rounds. Additionally, despite SPDnet's centralized score being lower than EEGnet's, FedSPDnet retains its centralized performance, indicating that geometry-aware aggregation provides intrinsic regularization. Ablation studies reveal that ProjAvg and RLAvg perform similarly across different datasets, validating their applicability in various scenarios.

FedSPDnet has broad application scenarios in signal processing applications, especially in scenarios requiring data privacy protection. Specifically, this method is suitable for motor imagery classification of EEG data, micro-Doppler radar signal processing, and ground-penetrating radar image classification. In these scenarios, SPD matrices serve as feature descriptors, effectively capturing the geometric structure of the data to improve classification accuracy. Additionally, FedSPDnet's geometry-aware aggregation strategies enable effective information sharing between different clients, enhancing the model's generalization capabilities.

Although FedSPDnet performs well on multiple datasets, it may encounter computational bottlenecks when handling extremely large datasets, particularly in calculating the Riemannian mean. ProjAvg and RLAvg strategies may require additional storage and computational resources in some cases, potentially limiting their application on resource-constrained devices. Additionally, while FedSPDnet performs well on EEG datasets, its performance on other types of datasets still needs further validation. Future research directions include further optimizing FedSPDnet's computational efficiency for application on larger datasets.