Averaging symmetric positive-definite matrices on the space of eigen-decompositions
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We study extensions of Fréchet means for random objects in the space Sym^+(p) of p × p symmetric positive-definite matrices using the scaling-rotation geometric framework introduced by Jung et al. [SIAM J. Matrix. Anal. Appl. 36 (2015) 1180-1201]. The scaling-rotation framework is designed to enjoy a clearer interpretation of the changes in random ellipsoids in terms of scaling and rotation. In this work, we formally define the scaling-rotation (SR) mean set to be the set of Fréchet means in Sym^+(p) with respect to the scaling-rotation distance. Since computing such means requires a difficult optimization, we also define the partial scaling-rotation (PSR) mean set lying on the space of eigen-decompositions as a proxy for the SR mean set. The PSR mean set is easier to compute and its projection to Sym^+(p) often coincides with SR mean set. Minimal conditions are required to ensure that the mean sets are non-empty. Because eigen-decompositions are never unique, neither are PSR means, but we give sufficient conditions for the sample PSR mean to be unique up to the action of a certain finite group. We also establish strong consistency of the sample PSR means as estimators of the population PSR mean set, and a central limit theorem. In an application to multivariate tensor-based morphometry, we demonstrate that a two-group test using the proposed PSR means can have greater power than the two-group test using the usual affine-invariant geometric framework for symmetric positive-definite matrices.
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