Remove logarithmic gaps in minimax sample complexity for matrix normal covariance estimation

Let $d_1,d_2\ge 2$ and let $\mathbb{S}_{++}^{d}$ denote the set of $d\times d$ real symmetric positive-definite matrices. Assume i.i.d. matrix-normal observations $X_1,\dots,X_n\in\mathbb{R}^{d_1\times d_2}$ with mean zero and

for unknown $\Sigma_1\in\mathbb{S}_{++}^{d_1}$ and $\Sigma_2\in\mathbb{S}_{++}^{d_2}$. This implies

The parameterization is non-identifiable up to reciprocal scaling: $(\Sigma_1,\Sigma_2)$ and $(c\Sigma_1,c^{-1}\Sigma_2)$, $c>0$, induce the same distribution.

For $A,B\in\mathbb{S}_{++}^{d}$, define the Fisher--Rao distance

and the Thompson distance

For pairs, use scale-invariant losses

For $m\in\{\mathrm{FR},\mathrm{Th}\}$, define minimax risk

where the infimum is over all measurable estimators based on $X_1,\dots,X_n$.