Electroencephalography (EEG) and magnetoencephalography (MEG) measure the brain signal from spatially-distributed electrodes. In order to detect event-related synchronization and desynchronization (ERS/ERD), which are utilized for brain-computer/machine interfaces (BCI/BMI), spatial filtering techniques are often used. Common spatial potential (CSP) filtering and its extensions which are the spatial filtering methods have been widely used for BCIs. CSP transforms brain signals that have a spatial and temporal index into vectors via a covariance representation. However, the variance-covariance structure is essentially different from the vector space, and not all the information can be transformed into an element of the vector structure. Grassmannian embedding methods, therefore, have been proposed to utilize the variance-covariance structure of variational patterns. In this paper, we propose a metric learning method to classify the brain signal utilizing the covariance structure. We embed the brain signal in the extended Grassmann manifold, and classify it on the manifold using the proposed metric. Due to this embedding, the pattern structure is fully utilized for the classification. We conducted an experiment using an open benchmark dataset and found that the proposed method exhibited a better performance than CSP and its extensions.

Most of the recent classification methods require tuning of the hyper-parameters, such as the kernel function parameter and the regularization parameter. Cross-validation or the leave-one-out method is often used for the tuning, however their computational costs are much higher than that of obtaining a classifier. Quadratically constrained maximum a posteriori (QCMAP) classifiers, which are based on the Bayes classification rule, do not have the regularization parameter, and exhibit higher classification accuracy than support vector machine (SVM). In this paper, we propose a multiple kernel learning (MKL) for QCMAP to tune the kernel parameter automatically and improve the classification performance. By introducing MKL, QCMAP has no parameter to be tuned. Experiments show that the proposed classifier has comparable or higher classification performance than conventional MKL classifiers.

We propose a kernel-based quadratic classification method based on kernel principal component analysis (KPCA). Subspace methods have been widely used for multiclass classification problems, and they have been extended by the kernel trick. However, there are large computational complexities for the subspace methods that use the kernel trick because the problems are defined in the space spanned by all of the training samples. To reduce the computational complexity of the subspace methods for multiclass classification problems, we extend Oja's averaged learning subspace method and apply a subset approximation of KPCA. We also propose an efficient method for selecting the basis vectors for this. Due to these extensions, for many problems, our classification method exhibits a higher classification accuracy with fewer basis vectors than does the support vector machine (SVM) or conventional subspace methods.