Global rate optimality of integral curve estimators in high order tensor models

Motivated by an application in neuroimaging, we consider the problem of establishing global minimax lower bound in a high order tensor model. In particular, the methodology we describe provides the global minimax bound for the integral curve estimator proposed in [O. Carmichael and L. Sakhanenko, Linear Algebra Appl., 473 (2015), pp. 377–403] under a semiparametric estimation setting. The theoretical results in this paper guarantee that the estimator used for tracing the complex fiber structure inside a live human brain obtained from high angular resolution diffusion imaging (HARDI) data is not only optimal locally but also optimal globally. The global minimax bound on the asymptotic risk of the estimators thus will provide a quantification of uncertainty for the estimation method in the whole domain of the imaging field. In addition to theoretical results, we also provide a detailed simulation study in order to find the optimal number of gradient directions for the imaging protocols, which we further illustrate with a real data analysis of a live human brain scan to showcase the uncertainty quantification of the estimation method in [O. Carmichael and L. Sakhanenko, Linear Algebra Appl., 473 (2015), pp. 377–403]. Furthermore, based on the global minimax bound, we propose a method for comparing the relative accuracy of several commonly used neuroimaging protocols in diffusion tensor imaging (DTI).