  In this paper, we develop a new classification method for manifold-valued
data in the framework of probabilistic learning vector quantization. In many
classification scenarios, the data can be naturally represented by symmetric
positive definite matrices, which are inherently points that live on a curved
Riemannian manifold. Due to the non-Euclidean geometry of Riemannian manifolds,
traditional Euclidean machine learning algorithms yield poor results on such
data. In this paper, we generalize the probabilistic learning vector
quantization algorithm for data points living on the manifold of symmetric
positive definite matrices equipped with Riemannian natural metric
(affine-invariant metric). By exploiting the induced Riemannian distance, we
derive the probabilistic learning Riemannian space quantization algorithm,
obtaining the learning rule through Riemannian gradient descent. Empirical
investigations on synthetic data, image data , and motor imagery EEG data
demonstrate the superior performance of the proposed method.
