Differential Geometry of the Fixed-Rank Core Covariance Manifold

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.