Quotient geometry of bounded or fixed rank correlation matrices
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914890134847488 |
|---|---|
| author | Chen, Hengchao |
| author_facet | Chen, Hengchao |
| contents | This paper studies the quotient geometry of bounded or fixed-rank correlation matrices. We establish a bijection between the set of bounded-rank correlation matrices and a quotient set of a spherical product manifold by an orthogonal group. We show that it forms an orbit space, whose stratification is determined by the rank of the matrices, and the principal stratum has a compatible Riemannian quotient manifold structure. We show that any minimizing geodesic in the orbit space has constant rank on the interior of the segment. We also develop efficient Riemannian optimization algorithms for computing the distance and weighted the Frechet mean in the orbit space. Moreover, we examine geometric properties of the quotient manifold, including horizontal and vertical spaces, Riemannian metric, injectivity radius, exponential and logarithmic map, curvature, gradient and Hessian. Finally, we apply our approach to a functional connectivity study using the Autism Brain Imaging Data Exchange. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03126 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quotient geometry of bounded or fixed rank correlation matrices Chen, Hengchao Metric Geometry Other Statistics 51F99, 15B99, 53A99, 65K10 This paper studies the quotient geometry of bounded or fixed-rank correlation matrices. We establish a bijection between the set of bounded-rank correlation matrices and a quotient set of a spherical product manifold by an orthogonal group. We show that it forms an orbit space, whose stratification is determined by the rank of the matrices, and the principal stratum has a compatible Riemannian quotient manifold structure. We show that any minimizing geodesic in the orbit space has constant rank on the interior of the segment. We also develop efficient Riemannian optimization algorithms for computing the distance and weighted the Frechet mean in the orbit space. Moreover, we examine geometric properties of the quotient manifold, including horizontal and vertical spaces, Riemannian metric, injectivity radius, exponential and logarithmic map, curvature, gradient and Hessian. Finally, we apply our approach to a functional connectivity study using the Autism Brain Imaging Data Exchange. |
| title | Quotient geometry of bounded or fixed rank correlation matrices |
| topic | Metric Geometry Other Statistics 51F99, 15B99, 53A99, 65K10 |
| url | https://arxiv.org/abs/2401.03126 |