Geometric mean for T-positive definite tensors and associated Riemannian geometry

초록

In this paper, we generalize the geometric mean of two positive definite matrices to that of third-order tensors using the notion of T-pro duct. Specifically, we define the geometric mean of two T-positive definite tensors and verify several properties that "mean" should satisfy including the idempotence and the commutative property, and so on. Moreover, it is shown that the geometric mean is a unique T-positive definite solution of an algebraic Riccati tensor equation and can be expressed as solutions of algebraic Riccati matrix equations. In addition, we investigate the Riemannian manifold associated with the geometric mean for T-positive definite tensors, considering it as a totally geodesic embedded submanifold of the Riemannian manifold associated with the case of matrices. It is particularly shown that the geometric mean of two T-positive definite tensors is the midpoint of a unique geodesic joining the tensors, and the manifold is a Cartan-Hadamard-Riemannian manifold. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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