Rotation-based metric on the Riemannian manifold of SPD matrices with applications to source data selection for brain-computer interface transfer learning

This paper introduces the pole ratio metric and presents a sphere-based view of symmetric positive-definite matrix rotations on the Riemannian manifold of symmetric positive-definite matrices equipped with the affine-invariant Riemannian metric. The pole ratio quantifies whether data from different users lie on this Riemannian manifold in a way that enables effective transfer learning. The sphere-based view provides insight into the rotational step of transfer learning using the Riemannian Procrustes analysis method and highlights the limitations of rotation. For effective transfer learning, selecting appropriate source data is essential for good performance. The pole ratio is shown to be an effective metric for selecting source data. The main contribution of the paper is the insight into the limitations of rotations on a Riemannian manifold; the usefulness of the pole ratio as a source selection metric is a natural extension of this insight. This paper focuses on Brain-Computer Interfaces (BCIs), but the sphere-based view of rotations of symmetric positive-definite matrix data and the pole ratio are applicable to any field that models two-class data using symmetric positive-definite matrices.