AI 中文总结
本文针对协方差为克罗内克积的高斯张量，将张量正态极大似然估计的样本阈值中$d_{\rm max}$的三次依赖改进为二次依赖，达到信息论最优，解决了前期工作提出的开放问题。
AI 中文摘要
设$X_1,\boldsymbol{\rmellipsis},X_n$为$\boldsymbol{\rmellipsis}^{d_1}\boldsymbol{\rmellipsis}\boldsymbol{\rmellipsis}\boldsymbol{\rmellipsis}^{d_k}$中的独立高斯张量，其协方差为$k$个未知正定因子的克罗内克积，记$D=\boldsymbol{\rmprod}_{a=1}^k d_a$，$d_{\rm max}=\boldsymbol{\rmmax}_a d_a$。Franks等人（2026）的最新结果在样本阈值$nD\boldsymbol{\rmgtrsim}k^2 d_{\rm max}^3$下，为张量正态极大似然估计量建立了无条件数的非渐近保证，并提出是否可将$d_{\rm max}$的三次依赖替换为算子范数尺度$d_{\rm max}^2$的问题。本文肯定回答了该问题，证明对于$t\boldsymbol{\rmgeq}1$，只要$nD\boldsymbol{\rmgeq}Ck^2 d_{\rm max}^2 t^2$，极大似然估计量以高概率唯一存在，且满足$d_{\rm FR}(\boldsymbol{\rmwidehat{\boldsymbol{\rmTheta}}},\boldsymbol{\rmTheta})\boldsymbol{\rmleq}Ct\boldsymbol{\rmsqrt{k}}\boldsymbol{\rmcdot}d_{\rm max}/\boldsymbol{\rmsqrt{n}}$，$d_{\rm FR}(\boldsymbol{\rmwidehat{\boldsymbol{\rmTheta}}}_a,\boldsymbol{\rmTheta}_a)\boldsymbol{\rmleq}Ct\boldsymbol{\rmsqrt{k d_a}}\boldsymbol{\rmcdot}d_{\rm max}/\boldsymbol{\rmsqrt{nD}}$；对每个最大维度模态，还得到尖锐Thompson界$d_{\rm op}(\boldsymbol{\rmwidehat{\boldsymbol{\rmTheta}}}_a,\boldsymbol{\rmTheta}_a)\boldsymbol{\rmleq}Ct\boldsymbol{\rmcdot}d_{\rm max}/\boldsymbol{\rmsqrt{nD}}$。本文未假设稀疏性、条件数界或 warm start，固定$k$时，该阈值对$d_{\rm max}$的依赖达到信息论最优，全精度与最大因子的估计速率与高斯极小极大下界仅差$\boldsymbol{\rmsqrt{k}}$因子。证明将局部群轨道方向的随机Gram界扩展至整个局部李代数，通过精确共轭将其转移至固定Thompson球，并结合约束极大似然估计量的敏感性、等变Kirszbraun延拓与高斯集中性，消除了导致之前额外因子$d_{\rm max}$的Frobenius-to-operator损失，解决了前期工作提出的明确开放问题。
英文摘要
Let $X_1,\ldots,X_n$ 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 $t\geq 1$, if $nD\geq C k^2 d_{\max}^2 t^2$, then with high probability the maximum likelihood estimator exists, is unique, and satisfies $d_{\rm FR}(\widehatΘ,Θ)\leq C t \sqrt{k} d_{\max}/\sqrt{n}$ and $d_{\rm FR}(\widehatΘ_a,Θ_a)\leq C t\sqrt{k d_a} d_{\max}/\sqrt{nD}$ for every mode $a$. For every mode $a$ with $d_a=d_{\max}$, we further establish the sharp Thompson-metric bound $d_{\rm op}(\widehatΘ_a,Θ_a)\leq C t d_{\max}/\sqrt{nD}$. 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.
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