Full minimax characterization for tensor normal factor estimation beyond the largest factor

Let $K\ge 3$ and dimensions $d_1,\dots,d_K\in\mathbb{N}$. For each sample, observe a random tensor $X\in\mathbb{R}^{d_1\times\cdots\times d_K}$, and write $\operatorname{vec}(X)\in\mathbb{R}^{D}$ with $D=\prod_{j=1}^K d_j$. Assume

for $i=1,\dots,n$, where $\mathbb{S}_{++}^{d}$ is the cone of $d\times d$ real symmetric positive-definite matrices and $\otimes$ is the Kronecker product.

Because the factorization is not unique under rescalings $\Sigma_k\mapsto c_k\Sigma_k$ with $c_k>0$ and $\prod_{k=1}^K c_k=1$, define identifiable factor parameters by shape normalization

For estimation error, use the affine-invariant Fisher-Rao distance

and optionally the Thompson/log-spectral form

These two metrics are comparable but not dimension-free equivalent: for $d\times d$ SPD matrices,

Define minimax risks

where infima are over all measurable estimators based on $X_1,\dots,X_n$, and the supremum is over all unrestricted positive-definite factors (no bounded condition number, sparsity, or structural constraints unless explicitly imposed).