A0258
Title: Averaging symmetric positive-definite matrices on the space of eigen-decompositions
Authors: Sungkyu Jung - Seoul National University (Korea, South) [presenting]
Abstract: Extensions of Fr\'{e}chet means for random objects in the space of symmetric positive-definite matrices are studied using the scaling-rotation geometric framework. The scaling-rotation framework is designed to enjoy a clearer interpretation of the changes in random ellipsoids in terms of scaling and rotation. The framework has been beneficial in smoothing coarse diffusion tensor imaging. We formally define the scaling-rotation (SR) mean set as the set of Fr\'{e}chet means with respect to the scaling-rotation distance. Since computing such means requires a discrete optimization, we instead define the partial scaling-rotation (PSR) mean set lying on the space of eigen-decompositions as a proxy for the SR mean set, which is easier to compute and often coincides with the SR mean set. Even though the PSR mean is never unique, we reveal sufficient conditions for the mean to be unique up to the action of a certain group. On a theoretical side, a procedure is illustrated for deriving strong consistency and a central limit theorem for M-estimators, defined in a non-metric and stratified space. In an application to multivariate tensor-based morphometry, we demonstrate that a two-group test using the proposed PSR means has greater power than using the usual Log-Euclidean geometric framework for symmetric positive-definite matrices.
