Covariance Estimation for Matrix-variate Data via Fixed-rank Core Covariance Geometry

We study the geometry of the fixed-rank core covariance manifold and propose a novel covariance estimator for matrix-variate data leveraging this geometry. To generalize the separable covariance model, Hoff, McCormack, and Zhang (2023) showed that 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 $p:=p_1p_2$. 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, whereas $\mathcal{C}{p_1,p_2}^{++}:=\mathcal{C}{p_1,p_2,p}^+$ 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. Using this geometry, we propose a partial-isotropy core shrinkage estimator for matrix-variate data, supported by numerical illustrations.

💡 Research Summary

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The paper investigates covariance estimation for matrix‑valued observations by exploiting the geometry of a fixed‑rank “core” covariance component. Building on the Kronecker‑core decomposition (KCD) introduced by Hoff, McCormack and Zhang (2023), any p₁ × p₂ covariance matrix Σ can be uniquely written as Σ = K^{1/2} C K^{1/2}ᵀ, where K is the most separable part (a Kronecker product of two positive‑definite matrices) and C is the core. When Σ has rank r and the dimensionality condition p₁/p₂ + p₂/p₁ < r holds, C also has rank r.

The authors address the open question of whether a “partial‑isotropy” structure can be imposed on C to aid high‑dimensional estimation. They show that a partial‑isotropy rank‑r core can be expressed as a non‑trivial convex combination of a rank‑r core and the identity matrix Iₚ: \