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

Abstract

Let $X_1,\ldots,X_n$ be independent Gaussian tensors in $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_k}$ whose covariance is a Kronecker product of $k$ unknown positive-definite factors, and put $D=\prod_{a=1}^k d_a$ and $d_{\max}=\max_a d_a$. A recent result of Franks et al. (2026) established condition-number-free nonasymptotic guarantees for the tensor-normal maximum likelihood estimator under the sample threshold $nD\gtrsim k^2 d_{\max}^3$. They asked whether the cubic dependence on $d_{\max}$ could be replaced by the operator-norm scale $d_{\max}^2$. We answer this question affirmatively. We prove that, for $t\geq 1$, the maximum likelihood estimator exists uniquely with high probability whenever $nD\geq Ck^2 d_{\max}^2 t^2$, and satisfies $d_{\mathrm{FR}}(\widehatΘ,Θ)\leq Ct\sqrt{k}\,d_{\max}/\sqrt{n}$ and $d_{\mathrm{FR}}(\widehatΘ_a,Θ_a)\leq Ct\sqrt{k d_a}\,d_{\max}/\sqrt{nD}$. For every mode of largest dimension, we also obtain the sharp Thompson bound $d_{\mathrm{op}}(\widehatΘ_a,Θ_a)\leq Ct\,d_{\max}/\sqrt{nD}$. No sparsity, condition-number bound or warm start is assumed. For fixed $k$, the threshold has the information-theoretically optimal dependence on $d_{\max}$, and the displayed rates for the full precision and the largest factor match Gaussian minimax lower bounds up to a factor $\sqrt{k}$. The proof extends a random Gram bound for local group-orbit directions to the full local Lie algebra, transports it to a fixed Thompson ball by exact conjugation, and combines sensitivity of a constrained maximum likelihood estimator with an equivariant Kirszbraun extension and Gaussian concentration. This removes the Frobenius-to-operator loss responsible for the previous extra factor $d_{\max}$ and resolves the explicit open problem posed in the earlier work.

Disclosure

“previous extra factor dmax and resolves the explicit open problem posed in the earlier work. The proof was developed with substantial assistance from GPT-5.6 and Claude Fable 5, and was manually checked by the authors. Some key words: geodesic convexity; Kronecker covariance; maximum likelihood; operator norm”

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Classification

Substantial proof generation

Multiplier

10

Verified

2026-08-23T05:09:00.546524Z

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