Tensor-normal maximum likelihood estimation at the operator-norm sample threshold

Abstract

Let $X_1,\dots ,X_n$ be independent Gaussian tensors in ${\mathbb{R}}^{d_1}\otimes \cdots \otimes {\mathbb{R}}^{d_k}$ with a common covariance matrix given by the Kronecker product of $k$ unknown positive-definite factors, and let $D={\prod }_{a=1}^k{d}_a$ and $d_{\max }={\max }_a{d}_a$. Franks et al. (2026) established condition-number-free guarantees for the tensor-normal maximum likelihood estimator under the sample-size condition $nD\gtrsim k^2{d}_{\max }^3$ and asked whether the cubic dependence on $d_{\max }$ could be reduced to a quadratic one. We answer this question affirmatively. For $t\ge 1$, if $nD\ge C{k}^2{d}_{\max }^2{t}^2$, then with high probability the maximum likelihood estimator exists, is unique, and satisfies $d_{\mathrm{FR}}(\hat{\Theta },\Theta )\le Ct\sqrt{k}{d}_{\max }/\sqrt{n}$ and $d_{\mathrm{FR}}({\hat{\Theta }}_a,{\Theta }_a)\le Ct\sqrt{k{d}_a}{d}_{\max }/\sqrt{nD}$ for every mode $a$. For every mode $a$ with $d_a=d_{\max }$, we further establish the sharp Thompson-metric bound $d_{\mathrm{op}}({\hat{\Theta }}_a,{\Theta }_a)\le Ct{d}_{\max }/\sqrt{nD}$. These guarantees are uniform over the unknown covariance factors and require neither condition-number bounds nor sparsity assumptions. Gaussian submodel lower bounds match the full and largest-factor Fisher--Rao rates up to a factor of $\sqrt{k}$ and the largest-factor Thompson rate up to universal constants. Consequently, for fixed $k$, the quadratic dependence of the sample-size threshold on $d_{\max }$ is optimal. GPT-5.6 Sol and Claude Fable 5 were used to assist with proof development, verification, and manuscript preparation.

Work in Progress