Kernel nonnegative matrix factorization for spectral EEG feature extraction

Nonnegative matrix factorization (NMF) seeks a decomposition of a nonnegative matrix X⩾0X⩾0 into a product of two nonnegative factor matrices U⩾0U⩾0 and V⩾0V⩾0, such that a discrepancy between XX and UV⊤UV⊤ is minimized. Assuming U=XWU=XW in the decomposition (for W⩾0W⩾0), kernel NMF (KNMF) is easily derived in the framework of least squares optimization. In this paper we make use of KNMF to extract discriminative spectral features from the time–frequency representation of electroencephalogram (EEG) data, which is an important task in EEG classification. Especially when KNMF with linear kernel is used, spectral features are easily computed by a matrix multiplication, while in the standard NMF multiplicative update should be performed repeatedly with the other factor matrix fixed, or the pseudo-inverse of a matrix is required. Moreover in KNMF with linear kernel, one can easily perform feature selection or data selection, because of its sparsity nature. Experiments on two EEG datasets in brain computer interface (BCI) competition indicate the useful behavior of our proposed methods.