Partial least squares regression on symmetric positive-definite matrices

Abstract

Recently there has been an increased interest in the analysis of differenttypes of manifold-valued data, which include data from symmetric positivedefinitematrices. In many studies of medical cerebral image analysis, amajor concern is establishing the association among a set of covariates andthe manifold-valued data, which are considered as responses for characterizingthe shapes of certain subcortical structures and the differences betweenthem.The manifold-valued data do not form a vector space, and thus, it is notadequate to apply classical statistical techniques directly, as certain operationson vector spaces are not defined in a general Riemannian manifold. Inthis article, an application of the partial least squares regression methodologyis performed for a setting with a large number of covariates in a euclideanspace and one or more responses in a curved manifold, called a Riemanniansymmetric space. To apply such a technique, the Riemannian exponentialmap and the Riemannian logarithmic map are used on a set of symmetricpositive-definite matrices, by which the data are transformed into a vectorspace, where classic statistical techniques can be applied. The methodologyis evaluated using a set of simulated data, and the behavior of the techniqueis analyzed with respect to the principal component regression.