Geometric mean for T-positive definite tensors and associated Riemannian geometry
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910392535482368 |
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| author | Ju, Jeong-Hoon Kim, Taehyeong Kim, Yeongrak Choi, Hayoung |
| author_facet | Ju, Jeong-Hoon Kim, Taehyeong Kim, Yeongrak Choi, Hayoung |
| contents | In this paper, we generalize the geometric mean of two positive definite matrices to that of third-order tensors using the notion of T-product. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00255 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric mean for T-positive definite tensors and associated Riemannian geometry Ju, Jeong-Hoon Kim, Taehyeong Kim, Yeongrak Choi, Hayoung Numerical Analysis Differential Geometry Functional Analysis 15A69, 15A72, 15B48, 47A64, 53A45 In this paper, we generalize the geometric mean of two positive definite matrices to that of third-order tensors using the notion of T-product. 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. |
| title | Geometric mean for T-positive definite tensors and associated Riemannian geometry |
| topic | Numerical Analysis Differential Geometry Functional Analysis 15A69, 15A72, 15B48, 47A64, 53A45 |
| url | https://arxiv.org/abs/2404.00255 |