Core Concepts

The geometric mean of two T-positive definite tensors is defined and shown to have properties that a "mean" should satisfy, including idempotence, inversion, commutativity, and transformation. The geometric mean is also characterized as the unique T-positive definite solution of an algebraic Riccati tensor equation, and can be expressed as solutions of algebraic Riccati matrix equations. Additionally, the Riemannian manifold associated with the geometric mean for T-positive definite tensors is investigated, and it is shown to be a Cartan-Hadamard-Riemannian manifold.

Abstract

The content starts by reviewing the notion of T-product and T-positive definite tensors, and their basic properties. It then defines the geometric mean of two T-positive definite tensors and proves that it satisfies various properties that a "mean" should have, such as idempotence, inversion, commutativity, and transformation.
The geometric mean is also shown to be the unique T-positive definite solution of an algebraic Riccati tensor equation, and can be expressed as solutions of algebraic Riccati matrix equations.
The content then introduces a Riemannian metric on the convex open cone of T-positive definite tensors, and interprets the geometric mean in terms of this Riemannian metric. It is proved that the geometric mean of two T-positive definite tensors is the midpoint of the geodesic joining the tensors, and that the Riemannian manifold is complete and has nonpositive curvature.

Stats

The content does not contain any explicit numerical data or statistics to support the key logics.

Quotes

The content does not contain any striking quotes supporting the key logics.

Key Insights Distilled From

by Jeong-Hoon J... at **arxiv.org** 04-02-2024

Deeper Inquiries

To generalize the geometric mean for T-positive definite tensors to the case of multiple tensors, we can extend the concept of the T-product to accommodate more than two tensors. By iteratively applying the T-product operation to a set of T-positive definite tensors, we can define the geometric mean for multiple tensors.
For example, given a set of T-positive definite tensors {A1, A2, ..., An}, the geometric mean for these tensors can be defined as the solution to the equation A#(A1, A2, ..., An) = A1 ∗ A2 ∗ ... ∗ An, where A# represents the geometric mean operation for multiple tensors. This generalization allows for the calculation of the central tensor that represents the "average" of the given set of tensors.

The Riemannian geometry associated with the geometric mean for T-positive definite tensors has several potential applications in various fields:
Data Clustering: The geometric mean can be used as a similarity measure between tensors, enabling the clustering of high-dimensional data based on their geometric relationships.
Image Processing: In image recognition and analysis, the Riemannian manifold associated with the geometric mean can be utilized to compare and classify image tensors, leading to improved image processing algorithms.
Machine Learning: The geometric mean can play a role in feature extraction and dimensionality reduction in machine learning tasks, enhancing the efficiency and accuracy of models.
Signal Processing: Applications in signal processing can benefit from the geometric mean for T-positive definite tensors to analyze and process complex signals efficiently.
Statistical Analysis: The geometric mean can be used in statistical analysis to compare and combine multiple tensors, providing insights into the underlying data distribution.

The concepts and results presented in this work can be extended to other tensor operations beyond the T-product by exploring the properties and applications of different tensor operations. Some possible extensions include:
Generalized Tensor Products: Investigating the geometric mean for tensors under different tensor product definitions, such as the Kronecker product or the Hadamard product, to explore their geometric properties and applications.
Tensor Decompositions: Extending the Riemannian geometry framework to tensor decomposition methods like CP decomposition or Tucker decomposition to analyze the geometric relationships between tensor factors.
Tensor Norms and Metrics: Studying the geometric mean in the context of different tensor norms and metrics to understand the geometric structures of tensors and their applications in various domains.
Tensor Optimization: Applying the geometric mean concept in tensor optimization problems to find optimal solutions and explore the convergence properties of iterative tensor algorithms.
By exploring these extensions, we can further enhance our understanding of tensor operations and their geometric interpretations in diverse fields.

0