Learning with Iwasawa Coordinates

Finding a good metric over the input space plays a fundamental role in machine learning. Most existing techniques assume the Mahalanobis metric without incorporating the geometry of Pn, the space of n×n symmetric positive-definite (SPD) matrices, which leads to difficulties in the optimization procedure used to learn the metric. In this paper, we introduce a novel algorithm to learn the Mahalanobis metric using a natural parametrization of Pn. The data are then transformed by the learned metric into another linear space. This linear space however does not have the required structure needed to significantly improve the classification of data that are not linearly separable. Therefore, we develop an efficient algorithm to map this transformed input data space onto the known curved space of positive definite matrices, Pn, and empirically show that this mapping yields superior clustering results (in comparison to state-of-the-art) on several well published data. A key advantage of mapping the data to Pn as opposed to an infinite dimensional space using a Kernel (as in SVM) is that, Pn is finite dimensional, its geometry is fully known and therefore one can incorporate its Riemannian structure into nonlinear learning tasks.