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

Let X1,ldots,Xn be independent Gaussian tensors in ℝd1otimescdotsotimesℝdk with a common covariance matrix given by the Kronecker product of k unknown positive-definite factors, and let D=proda=1k da and dmax=maxa da. Franks et al. (2026) established condition-number-free guarantees for the tensor-normal maximum likelihood estimator under the sample-size condition nDgtrsim k2 dmax3 and asked whether the cubic dependence on dmax could be reduced to a quadratic one. We answer this question affirmatively. For t≥ 1, if nD≥ C k2 dmax2 t2, then with high probability the maximum likelihood estimator exists, is unique, and satisfies drm FR(widehatTheta,Theta)≤ C t √k dmax/√n and drm FR(widehatThetaa,Thetaa)≤ C t√k da dmax/√nD for every mode a. For every mode a with da=dmax, we further establish the sharp Thompson-metric bound drm op(widehatThetaa,Thetaa)≤ C t dmax/√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 √k and the largest-factor Thompson rate up to universal constants. Consequently, for fixed k, the quadratic dependence of the sample-size threshold on dmax is optimal. GPT-5.6 Sol and Claude Fable 5 were used to assist with proof development, verification, and manuscript preparation.