Covariance Estimation for Matrix-variate Data via Fixed-rank Core Covariance Geometry
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910121682010112 |
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| author | Sung, Bongjung |
| author_facet | Sung, Bongjung |
| contents | We study the geometry of the fixed-rank core covariance manifold arising from the Kronecker-core decomposition of covariance matrices. As shown in Hoff, McCormack, and Zhang (2023), every covariance matrix $Σ$ of $p_1\times p_2$ matrix-variate data uniquely decomposes into a separable component $K$ and a core component $C$. Such a decomposition also exists for rank-$r$ $Σ$ if $p_1/p_2+p_2/p_1<r$, with $C$ sharing the same rank. If this core $C$ exhibits a partial-isotropy structure, then a partial-isotropy rank-$r$ core is a non-trivial convex combination of a rank-$r$ core and $I_p$ for $p:=p_1p_2$, where the weight on $I_p$ measures the deviation of $Σ$ from separability. This motivates studying the geometry of the space of rank-$r$ cores, $\mathcal{C}_{p_1,p_2,r}^+$. We show that $\mathcal{C}_{p_1,p_2,r}^+$ is a smooth manifold, except for a measure-zero subset associated with canonical decomposability. When $r=p$, $\mathcal{C}_{p_1,p_2}^{++}:=\mathcal{C}_{p_1,p_2,p}^+$ is itself a smooth manifold. The geometric properties, including smoothness of the positive definite cone via separability and the Riemannian gradient and Hessian operator relevant to $\mathcal{C}_{p_1,p_2,r}^+$, are also derived. As an application, we propose a partial-isotropy core shrinkage estimator for matrix-variate data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01070 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Covariance Estimation for Matrix-variate Data via Fixed-rank Core Covariance Geometry Sung, Bongjung Differential Geometry Methodology We study the geometry of the fixed-rank core covariance manifold arising from the Kronecker-core decomposition of covariance matrices. As shown in Hoff, McCormack, and Zhang (2023), every covariance matrix $Σ$ of $p_1\times p_2$ matrix-variate data uniquely decomposes into a separable component $K$ and a core component $C$. Such a decomposition also exists for rank-$r$ $Σ$ if $p_1/p_2+p_2/p_1<r$, with $C$ sharing the same rank. If this core $C$ exhibits a partial-isotropy structure, then a partial-isotropy rank-$r$ core is a non-trivial convex combination of a rank-$r$ core and $I_p$ for $p:=p_1p_2$, where the weight on $I_p$ measures the deviation of $Σ$ from separability. This motivates studying the geometry of the space of rank-$r$ cores, $\mathcal{C}_{p_1,p_2,r}^+$. We show that $\mathcal{C}_{p_1,p_2,r}^+$ is a smooth manifold, except for a measure-zero subset associated with canonical decomposability. When $r=p$, $\mathcal{C}_{p_1,p_2}^{++}:=\mathcal{C}_{p_1,p_2,p}^+$ is itself a smooth manifold. The geometric properties, including smoothness of the positive definite cone via separability and the Riemannian gradient and Hessian operator relevant to $\mathcal{C}_{p_1,p_2,r}^+$, are also derived. As an application, we propose a partial-isotropy core shrinkage estimator for matrix-variate data. |
| title | Covariance Estimation for Matrix-variate Data via Fixed-rank Core Covariance Geometry |
| topic | Differential Geometry Methodology |
| url | https://arxiv.org/abs/2512.01070 |