Pseudo Affine-Invariant Riemannian Metrics for Efficient Brain-Computer Interfaces

Abstract

Objective: The Riemannian geometry of symmetric positive definite (SPD) matrices endowed with the affine-invariant (AI) metric represents the current state of the art for the classification of brain-computer interface (BCI) data based on electroencephalography. However, its computational complexity grows cubically with the size of the input matrices. A novel approach is presented to reduce this computational load while preserving classification performance using the minimum distance to mean (MDM) classifier. Methods: A form of data preconditioning is proposed, mutating the Euclidean metric into a pseudo-AI metric for the SPD manifold. The computation of he distance and barycenter on the manifold can then be executed with quadratic complexity in matrix size. The claim is validated in a within-session cross-validation scenario on 16 open-access BCI databases related to the motor imagery paradigm (6 databases, 69 sessions) and the P300 paradigm (10 databases, 155 sessions). Results: Classification performance is preserved. Training the MDM classifier allowed a speed-up factor of up to about 4× for P300 data. For MI data, a simpler pre-conditioner allowed a speed-up factor of up to about 15×. Conclusion: The proposed approach is simple, yet it can substantially reduce the energy consumption, carbon footprint, and execution time of distancebased SPD machine learning. Significance: The gain in execution time increases with the dimension of the input matrices and can reach several orders of magnitude for methods that heavily perform distance and barycenter operations on the SPD manifold. Open-source code is made available to the community