Partial Least Squares Regression on Symmetric Positive-Definite Matrices

Recently there has been an increased interest in the analysis of different types of manifold-valued data, which include data from symmetric positive-definite matrices. In many studies of medical cerebral image analysis, a major concern is establishing the association among a set of covariates and the manifold-valued data, which are considered as responses for characterizing the shapes of certain subcortical structures and the differences between them. The manifold-valued data do not form a vector space, and thus, it is not adequate to apply classical statistical techniques directly, as certain operations on vector spaces are not defined in a general Riemannian manifold. In this article, an application of the partial least squares regression methodology is performed for a setting with a large number of covariates in a euclidean space and one or more responses in a curved manifold, called a Riemannian symmetric space. To apply such a technique, the Riemannian exponential map and the Riemannian logarithmic map are used on a set of symmetric positive-definite matrices, by which the data are transformed into a vector space, where classic statistical techniques can be applied. The methodology is evaluated using a set of simulated data, and the behavior of the technique is analyzed with respect to the principal component regression.