SPD Learn: A Geometric Deep Learning Python Library for Neural Decoding Through Trivialization

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.

Brain signal decoding is challenging because neural recordings are noisy and vary across individuals, while labeled data are often limited. Recent attention-based models on the symmetric positive definite (SPD) manifold have nevertheless achieved strong performance using covariance and connectivity representations, yet the contribution of learned token weighting remains unclear. We examine two representative architectures, MAtt (based on log-Euclidean geometry) and GBWAtt (based on generalized Bures--Wasserstein geometry), and find that their learned attention weights remain close to uniform after training. We relate this behavior to bounded similarity parameterizations that, under the original softmax scaling, limit attention-weight contrast. Moreover, replacing learned weights with uniform weights, throughout training and evaluation, has little effect on mean predictive performance while preserving each model's original aggregation geometry. Motivated by these findings, we introduce SPD-MetaFormer, an attention-free architecture built on uniformly weighted Fréchet aggregation under log-Euclidean geometry. Its backbone uses a geodesic residual to update a summary token and a shared spectral feedforward map to transform all tokens, followed by a learned weighted readout. Token states remain SPD-valued until tangent-space classification. Across three EEG benchmarks, SPD-MetaFormer achieves competitive results relative to published Euclidean and manifold baselines. Separate matched reproductions test learned versus uniform weighting within MAtt and GBWAtt. These results suggest that, in the short-sequence and limited-data regimes studied, carefully designed SPD architectures can provide a simpler and effective alternative to adaptive manifold attention.

Decoding brain activity from electroencephalography (EEG) is crucial for neuroscience and clinical applications. Among recent advances in deep learning for EEG, geometric learning stands out as its theoretical underpinnings on symmetric positive definite (SPD) allows revealing structural connectivity analysis in a physics-grounded manner. However, current SPD-based methods focus predominantly on statistical aggregation of EEGs, with frequency-specific synchronization and local topological structures of brain regions neglected. Given this, we propose RepSPD, a novel geometric deep learning (GDL)-based model. RepSPD implements a cross-attention mechanism on the Riemannian manifold to modulate the geometric attributes of SPD with graph-derived functional connectivity features. On top of this, we introduce a global bidirectional alignment strategy to reshape tangent-space embeddings, mitigating geometric distortions caused by curvature and thereby enhancing geometric consistency. Extensive experiments demonstrate that our proposed framework significantly outperforms existing EEG representation methods, exhibiting superior robustness and generalization capabilities.