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

2026-02-26

AI Summary

Existing neural networks for symmetric positive definite (SPD) matrices in neural decoding lack a unified framework and exhibit inconsistent handling of manifold constraints, hindering reproducibility and integration. This work proposes SPD Learn—a modular Python library that embeds Stiefel and SPD manifold constraints directly into network architectures through trivialized parameterization, enabling standard backpropagation and end-to-end training in Euclidean space. The approach ensures numerical stability, is fully compatible with PyTorch, and seamlessly integrates with mainstream brain–computer interface toolchains such as MOABB, Braindecode, Nilearn, and SKADA. By implementing representative SPDNet models, SPD Learn advances the standardization, reproducibility, and practical deployment of neural decoding research.

Technology Category

Cognitive Modeling & Cognitive Systems: Neural Spike CodingMachine Learning: Learning with ManifoldsNatural Language Processing: Learning & Optimization for NLP

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingEconomics, Online Markets and Human Computation: Social networks and social learning

Abstract

Implementations of symmetric positive definite (SPD) matrix-based neural networks for neural decoding remain fragmented across research codebases and Python packages. Existing implementations often employ ad hoc handling of manifold constraints and non-unified training setups, which hinders reproducibility and integration into modern deep-learning workflows. To address this gap, we introduce SPD Learn, a unified and modular Python package for geometric deep learning with SPD matrices. SPD Learn provides core SPD operators and neural-network layers, including numerically stable spectral operators, and enforces Stiefel/SPD constraints via trivialization-based parameterizations. This design enables standard backpropagation and optimization in unconstrained Euclidean spaces while producing manifold-constrained parameters by construction. The package also offers reference implementations of representative SPDNet-based models and interfaces with widely used brain computer interface/neuroimaging toolkits and modern machine-learning libraries (e.g., MOABB, Braindecode, Nilearn, and SKADA), facilitating reproducible benchmarking and practical deployment.

Problem

• SPD matrices
• neural decoding
• geometric deep learning
• reproducibility
• manifold constraints

Innovation

• SPD matrices
• geometric deep learning
• trivialization
• manifold constraints
• neural decoding