Generalized Learning Riemannian Space Quantization: a Case Study on Riemannian Manifold of SPD Matrices

Learning vector quantization is a simple and efficient classification method, enjoying great popularity. However, in many classification scenarios, such as electroencephalogram (EEG) classification, the input features are represented by symmetric positive-definite matrices that live in a curved manifold rather than vectors that live in the flat Euclidean space. In this paper, we propose a new classification method for data points that live in curved Riemannian manifolds in the framework of learning vector quantization. The proposed method alters generalized learning vector quantization with Euclidean distance to the one operating under the appropriate Riemannian metric. We instantiate the proposed method for the Riemannian manifold of symmetric positive-definite matrices equipped with Riemannian natural metric. Empirical investigations on synthetic data and real-world motor imagery EEG data demonstrates that the performance of the proposed generalized learning Riemannian space quantization can significantly outperform the Euclidean generalized learning vector quantization (GLVQ), generalized relevance learning vector quantization (GRLVQ), and generalized matrix learning vector quantization (GMLVQ). The proposed method also shows competitive performance to the state-of-theart methods on the EEG classification of motor imagery tasks.