New multivariable mean from nonlinear matrix equation associated to the harmonic mean
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911778306260992 |
|---|---|
| author | Kim, Sejong Mer, Vatsalkumar N. |
| author_facet | Kim, Sejong Mer, Vatsalkumar N. |
| contents | Various multivariable means have been defined for positive definite matrices, such as the Cartan mean, Wasserstein mean, and Rényi power mean. These multivariable means have corresponding matrix equations. In this paper, we consider the following non-linear matrix equation: $$ X = \left[ \sum_{i=1}^{n} w_{i} [ (1-t) X + t A_{i} ]^{-1} \right]^{-1}, $$ where $t \in (0,1]$. We prove that this equation has a unique solution and define a new mean, which we denote as $G_{t}(ω; \mathbb{A})$. We explore important properties of the mean $G_{t}(ω; \mathbb{A})$ including the relationship with matrix power mean, and show that the mean $G_{t}(ω; \mathbb{A})$ is monotone in the parameter $t$. Finally, we connect the mean $G_{t}(ω; \mathbb{A})$ to a barycenter for the log-determinant divergence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10526 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New multivariable mean from nonlinear matrix equation associated to the harmonic mean Kim, Sejong Mer, Vatsalkumar N. Functional Analysis 15A24, 15B48 Various multivariable means have been defined for positive definite matrices, such as the Cartan mean, Wasserstein mean, and Rényi power mean. These multivariable means have corresponding matrix equations. In this paper, we consider the following non-linear matrix equation: $$ X = \left[ \sum_{i=1}^{n} w_{i} [ (1-t) X + t A_{i} ]^{-1} \right]^{-1}, $$ where $t \in (0,1]$. We prove that this equation has a unique solution and define a new mean, which we denote as $G_{t}(ω; \mathbb{A})$. We explore important properties of the mean $G_{t}(ω; \mathbb{A})$ including the relationship with matrix power mean, and show that the mean $G_{t}(ω; \mathbb{A})$ is monotone in the parameter $t$. Finally, we connect the mean $G_{t}(ω; \mathbb{A})$ to a barycenter for the log-determinant divergence. |
| title | New multivariable mean from nonlinear matrix equation associated to the harmonic mean |
| topic | Functional Analysis 15A24, 15B48 |
| url | https://arxiv.org/abs/2402.10526 |