A closed-form unsupervised geometry-aware dimensionality reduction method in the Riemannian Manifold of SPD matrices

Riemannian geometry has been found accurate and robust for classifying multidimensional data, for instance, in brain-computer interfaces based on electroencephalography. Given a number of data points on the manifold of symmetric positive-definite matrices, it is often of interest to embed these points in a manifold of smaller dimension. This is necessary for large dimensions in order to preserve accuracy and useful in general to speed up computations. Geometry-aware methods try to accomplish this task while respecting as much as possible the geometry of the original data points. We provide a closed-form solution for this problem in a fully unsupervised setting. Through the analysis of three brain-computer interface data bases we show that our method allows substantial dimensionality reduction without affecting the classification accuracy.