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Rotation-based metric on the Riemannian manifold of SPD matrices with applications to source data selection for

Frida Heskebeck1, Bo Bernhardsson1, Carolina Bergeling2

  • 1Department of Automatic Control, Lund University, Lund, Sweden.

Frontiers in Human Neuroscience
|June 8, 2026
PubMed
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The new pole ratio metric aids transfer learning by assessing symmetric positive-definite matrix data on Riemannian manifolds. This metric helps select optimal source data for improved performance in Brain-Computer Interfaces (BCIs) and other fields.

Area of Science:

  • Mathematics
  • Machine Learning
  • Computational Neuroscience

Background:

  • Transfer learning effectiveness relies on source data quality.
  • Symmetric positive-definite (SPD) matrices are used to model complex data, including in Brain-Computer Interfaces (BCIs).
  • Rotations of SPD matrices on Riemannian manifolds are crucial for transfer learning but have limitations.

Purpose of the Study:

  • Introduce the pole ratio metric for quantifying SPD matrix data distribution on Riemannian manifolds.
  • Provide a sphere-based view of SPD matrix rotations to understand transfer learning limitations.
  • Demonstrate the pole ratio's utility in selecting optimal source data for transfer learning.

Main Methods:

  • Developed a sphere-based geometric interpretation of SPD matrix rotations.
Keywords:
BCIRiemann geometrypole ratiosource data selectiontransfer learning

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  • Introduced the pole ratio metric derived from this geometric view.
  • Applied Riemannian Procrustes analysis to analyze rotational steps in transfer learning.
  • Evaluated the pole ratio's effectiveness in source data selection for transfer learning.

    • Main Results:

    • The pole ratio quantifies data distribution on the Riemannian manifold, indicating transfer learning potential.
    • The sphere-based view reveals inherent limitations in rotation-based transfer learning.
    • The pole ratio metric effectively identifies suitable source data, enhancing transfer learning performance.
    • The study highlights the importance of source data selection for successful transfer learning.

    Conclusions:

    • The pole ratio is a valuable metric for source data selection in transfer learning involving SPD matrices.
    • Understanding the geometric limitations of rotations on Riemannian manifolds is key for effective transfer learning.
    • The presented framework and pole ratio metric have broad applicability beyond BCIs to any field using SPD matrices for two-class data modeling.