Is Riemannian Geometry Better than Euclidean in Averaging Covariance Matrices for CSP-based Subject-Independent Classification of Motor Imagery?

Abstract

Common Spatial Pattern (CSP) is a popular feature extraction algorithm used for electroencephalogram (EEG) data classification in brain-computer interfaces. One of the critical operations used in CSP is taking the average of trial covariance matrices for each class. In this regard, the arithmetic mean, which minimizes the sum of squared Euclidean distances to the data points, is conventionally used; however, this operation ignores the Riemannian geometry in the manifold of covariance matrices. To alleviate this problem, Fréchet mean determined using different Riemannian distances have been used. In this paper, we are primarily concerned with the following question: Does using the Fréchet mean with Riemannian distances instead of arithmetic mean in averaging CSP covariance matrices improve the subject-independent classification of motor imagery (MI)? To answer this question we conduct a comparative study using the largest MI dataset to date, with 54 subjects and a total of 21,600 trials of left-and right-hand MI. The results indicate a general trend of having a statistically significant better performance when the Riemannian geometry is used.