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

Bruno Aristimunha, Ce Ju, Antoine Collas, Florent Bouchard, Ammar Mian, Bertrand Thirion, Sylvain Chevallier, Reinmar Kobler

TL;DR

SPD Learn is introduced, a unified and modular Python package for geometric deep learning with SPD matrices that provides core SPD operators and neural-network layers, including numerically stable spectral operators, and enforces Stiefel/SPD constraints via trivialization-based parameterizations.

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.

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

TL;DR

SPD Learn is introduced, a unified and modular Python package for geometric deep learning with SPD matrices that provides core SPD operators and neural-network layers, including numerically stable spectral operators, and enforces Stiefel/SPD constraints via trivialization-based parameterizations.

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.
Paper Structure (14 sections, 1 equation, 1 figure, 1 table)

This paper contains 14 sections, 1 equation, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Design Principles Overview
  3. Trivialization
  4. Tutorials for Neural Decoding Tasks
  5. Conclusions
  6. Functional
  7. Eigenvalue forward operator.
  8. Eigenvalue backward operator.
  9. Derived spectral operators.
  10. Module
  11. Manifold-Valued Parameters
  12. Stiefel manifold-valued parameters.
  13. SPD manifold-valued parameters.
  14. Positive definite scalar parameters.

Figures (1)

  • Figure 1: Literature Map: The x-axis denotes the released/publication date (more recent papers to the right) and the y-axis denotes citation count (more highly cited papers at the top). Directed edges indicate influence relationships, defined by citation links between papers. The map was generated with https://www.litmaps.com/ using data available as of January 2026.