Nonparametric Two-Sample Tests on Homogeneous Riemannian Manifolds, Cholesky Decompositions and Dyslexia Detection from Diffusion Tensor Imaging Outputs

This paper addresses much needed asymptotic and nonparametric bootstrap methodology for two-sample tests for means on Riemannian manifolds with a simply transitive group of isometries. In particular, we develop a two-sample procedure for testing the equality of the generalized Frobenius means of two independent populations on the space of symmetric positive matrices. The new method naturally lead to an analysis based on Cholesky decompositions of covariance matrices which helps to decrease computational time and does not increase dimensionality. The resulting nonparametric matrix valued statistics are used for testing if there is a difference on average between corresponding signals in Diffusion Tensor Images (DTI) in young children with dyslexia when compared to their clinically normal peers, based on data that was previously analyzed using parametric methods. Both methodologies show a significant difference. Short title: Two Sample Tests on Homogeneous Spaces