We study the differential geometry of the fixed-rank core covariance manifold. According to Hoff, McCormack, and Zhang [J. R. Stat. Soc., B: Stat., 85 (2023), pp. 1659--1679], 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. They posed an open question on whether a partial-isotropy structure can be imposed on $C$ for high-dimensional covariance estimation. We address this question by showing that a partial-isotropy rank-$r$ core is a non-trivial convex combination of a rank-$r$ core and $I_p$ for $p:=p_1p_2$, motivating the study of rank-$r$ cores. For fixed $r>p_1/p_2+p_2/p_1$, we prove that the set of rank-$r$ cores, $\mathcal{C}_{p_1,p_2,r}^+$, is a compact, smooth, embedded submanifold of the set of rank-$r$ positive semi-definite matrices, except for a measure-zero subset associated with canonical decomposability. When $r=p$, the set of full-rank cores $\mathcal{C}_{p_1,p_2}^{++}$ is itself a smooth manifold. Moreover, the positive definite cone $\mathcal{S}_p^{++}$ is diffeomorphic to the product of the Kronecker and core covariance manifolds, providing new geometric insight into $\mathcal{S}_p^{++}$ via separability. Differential geometric quantities, such as the differential of the diffeomorphism, as well as the Riemannian gradient and Hessian operator on $\mathcal{C}_{p_1,p_2}^{++}$ and the manifolds used in constructing $\mathcal{C}_{p_1,p_2,r}^+$, are also derived. Lastly, we propose a partial-isotropy core shrinkage estimator for matrix-variate data, supported by numerical illustrations.

翻译：我们研究了固定秩核心协方差流形的微分几何。根据Hoff、McCormack和Zhang的研究[J. R. Stat. Soc., B: Stat., 85 (2023), pp. 1659--1679]，每个$p_1\times p_2$矩阵变量数据的协方差矩阵$Σ$可唯一分解为可分离分量$K$和核心分量$C$。当$p_1/p_2+p_2/p_1<r$时，秩为$r$的$Σ$也存在此类分解，且$C$保持相同秩。他们提出了一个开放性问题：在高维协方差估计中，是否可以对$C$施加部分各向同性结构？我们通过证明部分各向同性秩-$r$核心是秩-$r$核心与$I_p$（其中$p:=p_1p_2$）的非平凡凸组合来回应此问题，从而推动了对秩-$r$核心的研究。对于固定的$r>p_1/p_2+p_2/p_1$，我们证明秩-$r$核心的集合$\mathcal{C}_{p_1,p_2,r}^+$是秩-$r$半正定矩阵集合的紧致、光滑、嵌入子流形，仅排除与典型可分解性相关的测度零子集。当$r=p$时，满秩核心集合$\mathcal{C}_{p_1,p_2}^{++}$本身构成光滑流形。此外，正定锥$\mathcal{S}_p^{++}$与Kronecker协方差流形及核心协方差流形的乘积微分同胚，这通过可分离性为$\mathcal{S}_p^{++}$提供了新的几何视角。我们还推导了微分几何量，包括微分同胚的微分，以及$\mathcal{C}_{p_1,p_2}^{++}$和用于构造$\mathcal{C}_{p_1,p_2,r}^+$的流形上的黎曼梯度和Hessian算子。最后，我们提出了一种针对矩阵变量数据的部分各向同性核心收缩估计器，并辅以数值示例加以验证。