Selective Cross-Subject Transfer Learning Based on Riemannian Tangent Space for Motor Imagery Brain-Computer Interface

A motor imagery (MI) brain-computer interface (BCI) plays an important role in the neurological rehabilitation training for stroke patients. Electroencephalogram (EEG)-based MI BCI has high temporal resolution, which is convenient for real-time BCI control. Therefore, we focus on EEG-based MI BCI in this paper. The identification of MI EEG signals is always quite challenging. Due to high inter-session/subject variability, each subject should spend long and tedious calibration time in collecting amounts of labeled samples for a subject-specific model. To cope with this problem, we present a supervised selective cross-subject transfer learning (sSCSTL) approach which simultaneously makes use of the labeled samples from target and source subjects based on Riemannian tangent space. Since the covariance matrices representing the multi-channel EEG signals belong to the smooth Riemannian manifold, we perform the Riemannian alignment to make the covariance matrices from different subjects close to each other. Then, all aligned covariance matrices are converted into the Riemannian tangent space features to train a classifier in the Euclidean space. To investigate the role of unlabeled samples, we further propose semi-supervised and unsupervised versions which utilize the total samples and unlabeled samples from target subject, respectively. Sequential forward floating search (SFFS) method is executed for source selection. All our proposed algorithms transfer the labeled samples from most suitable source subjects into the feature space of target subject. Experimental results on two publicly available MI datasets demonstrated that our algorithms outperformed several state-of-the-art algorithms using small number of the labeled samples from target subject, especially for good target subjects.

Manifolds

We now embark on the full theory, beginning with the concept of a manifold in differential geometry. The meaning of coordinates and coordinate transformations is carefully explained. The metric and its transformation between coordinate frames is discussed. Riemann normal coordinates are described. The concepts of a tangent space and local flatness are discussed and derived. It is shown how to use the metric to calculate distances, areas and volumes, and to describe submanifolds.

Obstruction flat rigidity of the CR 3-sphere

We consider the obstruction flatness problem for small deformations of the standard CR 3-sphere. That rigidity holds for the CR sphere was previously known (in all dimensions) for the case of embeddable CR structures, where it also holds at the infinitesimal level. In the 3-dimensional case, however, a CR structure need not be embeddable. Unlike in the embeddable case, it turns out that in the nonembeddable case there is an infinite-dimensional space of solutions to the linearized obstruction flatness equation on the standard CR 3-sphere and this space defines a natural complement to the tangent space of the embeddable deformations. In spite of this, we show that the CR 3-sphere does not admit nontrivial obstruction flat deformations, embeddable or nonembeddable.

Embedding Tangent Space Extreme Learning Machine for EEG Decoding in Brain Computer Interface Systems

In motor imagery brain computer interface system, the spatial covariance matrices of EEG signals which carried important discriminative information have been well used to improve the decoding performance of motor imagery. However, the covariance matrices often suffer from the problem of high dimensionality, which leads to a high computational cost and overfitting. These problems directly limit the application ability and work efficiency of the BCI system. To improve these problems and enhance the performance of the BCI system, in this study, we propose a novel semisupervised locality-preserving graph embedding model to learn a low-dimensional embedding. This approach enables a low-dimensional embedding to capture more discriminant information for classification by efficiently incorporating information from testing and training data into a Riemannian graph. Furthermore, we obtain an efficient classification algorithm using an extreme learning machine (ELM) classifier developed on the tangent space of a learned embedding. Experimental results show that our proposed approach achieves higher classification performance than benchmark methods on various datasets, including the BCI Competition IIa dataset and in-house BCI datasets.