---
格式版本: 2
标题: "Tensor-normal maximum likelihood estimation at the operator-norm sample threshold"
原文链接: "https://arxiv.org/abs/2608.10488"
发布日期: "2026-08-11"
发布时间校准状态: "found"
发布时间需复核: "否"
发布时间来源: "rule:local:strict_original_body"
发布时间证据: "**\\[v1\\]** Tue, 11 Aug 2026 05:03:38 UTC (48 KB)"
发布时间校准原因: "规则确认唯一严格发布时间，来源 local:strict_original_body"
发布时间校准置信度: "high"
发布时间候选数量: 18
发布时间严格候选数量: 6
发布时间原页读取状态: "source template page reused from URL open"
发布时间未找到原因: ""
发布时间校准时间: "2026-08-12T10:48:50+08:00"
发布时间仲裁状态: "skipped"
发布时间仲裁尝试次数: 0
发布时间仲裁耗时毫秒: 0
发现时间: "2026-08-12T10:46:47+08:00"
入库时间: "2026-08-12T02:48:50.828Z"
来源平台: "arXiv 学术论文搜索"
搜索渠道: "source_template"
搜索词: "https://arxiv.org/search/?query=Scale-up&searchtype=all"
匹配关键词:
  - "Scale-up"
相关厂家:
  []
相关专家:
  []
内容类型: "网页"
抓取工具: "Free Fetch + Defuddle"
清洗工具: "Defuddle Markdown + Defuddle/Readability 正文提取"
原始附件:
  []
AI优质: "否"
AI打分: 5
AI分档: "非优质"
AI质检状态: "不通过"
AI打分理由: "该论文为数学统计理论，与超节点/AI Rack/机柜级AI基础设施完全无关，命中关键词仅为搜索词，不提供任何相关技术或商业信息。"
AI质检模型: "ali-deepseek-v4-flash"
AI质检时间: "2026-08-12T10:48:54+08:00"
AI主题相关性: 0
AI来源权威性: 5
AI新颖性: 0
AI技术细节: 0
AI商业部署信号: 0
AI完整性: 0
采集批次: "2026年8月12日2点04分20秒"
采集批次ID: "20260812-020420-289"
去重键: "https://arxiv.org/abs/2608.10488"
---

## Mathematics > Statistics Theory

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

Authors:[Hengzhi He](https://arxiv.org/search/math?searchtype=author&query=He,+H), [Guang Cheng](https://arxiv.org/search/math?searchtype=author&query=Cheng,+G)

[View PDF](https://arxiv.org/pdf/2608.10488) [HTML (experimental)](https://arxiv.org/html/2608.10488v1)

> 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 $d_{\mathrm{FR}}(\widehat{\Theta},\Theta)\leq Ct\sqrt{k}\,d_{\max}/\sqrt{n}$ and $d_{\mathrm{FR}}(\widehat{\Theta}_a,\Theta_a)\leq Ct\sqrt{k d_a}\,d_{\max}/\sqrt{nD}$. For every mode of largest dimension, we also obtain the sharp Thompson bound $d_{\mathrm{op}}(\widehat{\Theta}_a,\Theta_a)\leq Ct\,d_{\max}/\sqrt{nD}$. 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.

| Comments: |  |
| --- | --- |
| Subjects: | Statistics Theory (math.ST); Machine Learning (stat.ML) |
| Cite as: | [arXiv:2608.10488](https://arxiv.org/abs/2608.10488) \[math.ST\] |
|  | (or [arXiv:2608.10488v1](https://arxiv.org/abs/2608.10488v1) \[math.ST\] for this version) |
|  | [https://doi.org/10.48550/arXiv.2608.10488](https://doi.org/10.48550/arXiv.2608.10488) |

## Submission history

From: Hengzhi He \[[view email](https://arxiv.org/show-email/0baa4858/2608.10488)\]  
**\[v1\]** Tue, 11 Aug 2026 05:03:38 UTC (48 KB)

[Which authors of this paper are endorsers?](https://arxiv.org/auth/show-endorsers/2608.10488) | Disable MathJax ([What is MathJax?](https://info.arxiv.org/help/mathjax.html))
