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

Jeong-Hoon Ju, Taehyeong Kim, Yeongrak Kim, Hayoung Choi

TL;DR

This work extends the classical geometric mean from positive definite matrices to third-order tensors using the T-product, defining $\mathcal{A} \# \mathcal{B}$ as $\mathcal{A}^{1/2} * (\mathcal{A}^{-1/2} * \mathcal{B} * \mathcal{A}^{-1/2})^{1/2} * \mathcal{A}^{1/2}$ and proving it remains T-positive definite. It shows the mean is the unique solution of the Riccati tensor equation $\mathcal{X} * \mathcal{A}^{-1} * \mathcal{X} = \mathcal{B}$ and can be computed via block-circulant diagonalization, connecting to the scalar matrix case through $A_i$ and $B_i$. The paper then develops a Riemannian geometry on the cone of T-positive definite tensors, proving that the induced geodesic distance makes $\mathcal{A} \# \mathcal{B}$ the midpoint of the unique geodesic between $\mathcal{A}$ and $\mathcal{B}$, and establishing that this tensor manifold is complete with nonpositive curvature (Cartan–Hadamard). An isometric embedding via $\mathrm{bcirc}$ ties the tensor geometry to the classical matrix setting, enabling analytic and interpolation interpretations. Overall, the results provide a rigorous framework for tensor interpolation and metric geometry with potential applications in multi-tensor data analysis and related fields.

Abstract

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.

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

TL;DR

This work extends the classical geometric mean from positive definite matrices to third-order tensors using the T-product, defining as and proving it remains T-positive definite. It shows the mean is the unique solution of the Riccati tensor equation and can be computed via block-circulant diagonalization, connecting to the scalar matrix case through and . The paper then develops a Riemannian geometry on the cone of T-positive definite tensors, proving that the induced geodesic distance makes the midpoint of the unique geodesic between and , and establishing that this tensor manifold is complete with nonpositive curvature (Cartan–Hadamard). An isometric embedding via ties the tensor geometry to the classical matrix setting, enabling analytic and interpolation interpretations. Overall, the results provide a rigorous framework for tensor interpolation and metric geometry with potential applications in multi-tensor data analysis and related fields.

Abstract

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.
Paper Structure (13 sections, 36 theorems, 85 equations)

This paper contains 13 sections, 36 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. T-product
  4. T-positive definite tensors
  5. T-eigenvalue decompositions
  6. Geometric mean of two T-positive definite tensors
  7. Definition and basic properties
  8. T-Löwner order
  9. Riemannian geometry
  10. Riemannian metric and geodesic
  11. Infinitesimal exponential metric increasing property
  12. Completeness and curvature
  13. Final remarks

Key Result

Lemma 2.1

Let $\mathcal{A} \in \mathbb{C}^{m \times n \times p}, \mathcal{B} \in \mathbb{C}^{n \times s \times p}$ and $\mathcal{C} \in \mathbb{C}^{s \times r \times p}$. Then

Theorems & Definitions (69)

  • Definition 2.1
  • Definition 2.2: kilmer2011factorization
  • Lemma 2.1: kilmer2011factorizationmiao2021t
  • Definition 2.3: kilmer2011factorization
  • Definition 2.4: kilmer2011factorization
  • Lemma 2.2
  • Lemma 2.3: kilmer2011factorization
  • Definition 2.5: kilmer2011factorization
  • Lemma 2.4: kilmer2011factorization
  • Definition 2.6: kilmer2011factorization
  • ...and 59 more