Multiclass Brain–Computer Interface Classification by Riemannian Geometry

Abstract

This paper presents a new classification framework for brain-computer interface (BCI) based on motor imagery. This framework involves the concept of Riemannian geometry in the manifold of covariance matrices. The main idea is to use spatial covariance matrices as EEG signal descriptors and to rely on Riemannian geometry to directly classify these matrices using the topology of the manifold of symmetric and positive definite (SPD) matrices. This framework allows to extract the spatial information contained in EEG signals without using spatial filtering. Two methods are proposed and compared with a reference method [multiclass Common Spatial Pattern (CSP) and Linear Discriminant Analysis (LDA)] on the multiclass dataset IIa from the BCI Competition IV. The first method, named minimum distance to Riemannian mean (MDRM), is an implementation of the minimum distance to mean (MDM) classification algorithm using Riemannian distance and Riemannian mean. This simple method shows comparable results with the reference method. The second method, named tangent space LDA (TSLDA), maps the covariance matrices onto the Riemannian tangent space where matrices can be vectorized and treated as Euclidean objects. Then, a variable selection procedure is applied in order to decrease dimensionality and a classification by LDA is performed. This latter method outperforms the reference method increasing the mean classification accuracy from 65.1% to 70.2%.

Riemannian Laplace Distribution on the Space of Symmetric Positive Definite Matrices

“…Recently, several approaches have used the Riemannian distance in general as the main innovation in image or signal classification problems [2,15,34]. It turns out that the use of this distance leads to more accurate results (in comparison, for example, with the Euclidean distance).…”

“…When choosing between two clusters with the same number of points and the same dispersion, this rule favors the one whose median is closer to Y t . If the number of data points inside clusters and the respective dispersions are neglected, then Equation (36) reduces to the nearest neighbor rule involving only the Riemannian distance introduced in [2].…”

“…Data with values in the space P m , of m × m symmetric positive definite matrices, play an essential role in many applications, including medical imaging [1,2], computer vision [3][4][5][6][7] and radar signal processing [8,9]. In these applications, the location where a dataset is centered has a special interest.…”

“…For the Gaussian distribution Equation (2), the maximum likelihood estimate (MLE) for the parameterȲ based on observations Y 1 , · · · , Y n corresponds to the mean Equation (1). In [15], a detailed study of statistical inference for this distribution was given and then applied to the classification of data in P m , showing that it yields better performance, in comparison to recent approaches [2].…”

“…In [15], a detailed study of statistical inference for this distribution was given and then applied to the classification of data in P m , showing that it yields better performance, in comparison to recent approaches [2].…”