Kernel learning over the manifold of symmetric positive definite matrices for dimensionality reduction in a BCI application

Abstract

In this paper, we propose a kernel for nonlinear dimensionality reduction over the manifold of Symmetric Positive Definite (SPD) matrices in a Motor Imagery (MI)-based Brain Computer Interface (BCI) application. The proposed kernel, which is based on Riemannian geometry, tries to preserve the topology of data points in the feature space. Topology preservation is the main challenge in nonlinear dimensionality reduction (NLDR). Our main idea is to decrease the non-Euclidean characteristics of the manifold by modifying the volume elements. We apply a conformal transform over data-dependent isometric mapping to reduce the negative eigen fraction to learn a data dependent kernel over the Riemannian manifolds. Multiple experiments were carried out using the proposed kernel for a dimensionality reduction of SPD matrices that describe the EEG signals of dataset IIa from BCI competition IV. The experiments show that this kernel adapts to the input data and leads to promising results in comparison with the most popular manifold learning methods and the Common Spatial Pattern (CSP) technique as a reference algorithm in BCI competitions. The proposed kernel is strong, particularly in the cases where data points have a complex and nonlinear separable distribution.

Motor Imagery EEG Classification Based on Decision Tree Framework and Riemannian Geometry

“…Table 5 displays the comparison of classification accuracy using SJGDA and KNN for L/R task in 10-folder cross validation. References [8, 43–45] contain the classification of other publications. We have improved the accuracy compared with Reference [44] ( p = 0.85) and Reference [45] ( p =0.45).…”

“…Common spatial pattern (CSP) is widely used in motor imagery to extract EEG features [7]. CSP has excellent performance in two classification tasks, but the drawback is that it needs a lot of electrodes [8].…”

“…Lee et al used discrete wavelet transform (DWT) and continuous wavelet transform (CWT) to extract features of MI tasks, and Gaussian mixture model (GMM) was used to construct GMM supervectors; this method accelerates the speed of training and improves the accuracy of motor imagery [26]. Sadatnejad et al propose a new kernel to preserve the topology of data points in the feature space, and the proposed kernel is strong, particularly in the cases where data points have a complex and nonlinear separable distribution [8]. Xie et al proposed a framework for intrinsic submanifold learning from a high-dimensional Riemannian manifold; the proposed method exhibited strong robustness against a small training dataset [27].…”

Riemannian geometry for EEG-based brain-computer interfaces; a primer and a review

“…In parallel, geometry-aware dimensionality reduction (see section 3.1) inspired by the Riemannian framework is currently intensely investigated. Works related to BCI data include [126][127][128][129][130][131]. A relevant work, which can be readily borrowed from the computer vision community, is [132,133].…”

Cognitive facial expression recognition with constrained dimensionality reduction

“…space. To consider these problems together, some dimensionality reduction algorithms are proposed [17][18][19][20][21]. Among them, the semi-supervised long-term relevance feedback (RF) algorithm is much powerful [21], which can present the excellent representation for multimedia data.…”