发表机构
Marshall School of Business, University of Southern California; School of Public Health, University of California, Berkeley（南加州大学马歇尔商学院; 加州大学伯克利分校公共卫生学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对无约束协方差下高维矩形的高斯近似问题，通过两阶段插值与无秩矩阵加权高斯曲面界的方法，否定了Chernozhukov等人关于多项式维度下$n^{-1/4}$速率接近最优的推测，得出新的近似误差界为$n^{-1/3}$。
AI 中文摘要
针对无约束协方差下高维矩形的高斯近似问题，Chernozhukov等人（2023b）推测，在对数因子范围内，$n^{-1/4}$的速率接近最优。我们证明，在Chernozhukov等人（2023b）提出的尺度为$B_n$的逐坐标次指数条件和常数为$b$的边际方差下界条件下，维度$d$下的近似误差被$C_b\boldsymbol{\times}\boldsymbol{\text{min}}\boldsymbol{\bigg\bracerace{1, \bigg(\frac{B_n^2}{n}\bigg)^{1/3}\boldsymbol{\bigg\bracerace{log(2dn)\bigg\rbrace}^{7/3} + \frac{B_n}{\boldsymbol{\text{sqrt}}\boldsymbol{n}}\bigg\bracerace{log(2dn)\bigg\rbrace}^{5/2}\bigg\rbrace}$界定。特别地，对于有界$B_n$和多项式维度，新的界为$n^{-1/3}$，因此否定了多项式维度下$n^{-1/4}$接近最优的推测。该证明采用了两阶段插值和无秩矩阵加权高斯曲面界，可能具有独立研究价值。初始证明尝试由ChatGPT 5.6 Pro（OpenAI）生成，随后由作者修正并重写，证明的机器验证Lean形式化可在GitHub仓库（此https URL）获取。
英文摘要
For Gaussian approximation over high-dimensional rectangles under unrestricted covariance, Chernozhukov et al. (2023b) conjectured that the $n^{-1/4}$ rate, up to logarithmic factors, is near-optimal. We show that, under the coordinatewise subexponential condition with scale $B_n$ and the marginal variance lower bound condition with constant $b$ in Chernozhukov et al. (2023b), the approximation error in dimension $d$ is bounded by \begin{align*} C_b\min\left\{ 1,\, \left(\frac{B_n^2}{n}\right)^{1/3}\{\log(2dn)\}^{7/3} + \frac{B_n}{\sqrt n}\{\log(2dn)\}^{5/2} \right\}. \end{align*} In particular, for bounded $B_n$ and polynomial dimension, the new bound is $n^{-1/3}$ and therefore falsifies the polynomial-dimensional $n^{-1/4}$ near-optimality conjecture. The proof uses a two-stage interpolation and a rank-free matrix-weighted Gaussian surface bound, which may be of independent interest. The initial proof attempt was generated by ChatGPT 5.6 Pro (OpenAI) and subsequently corrected and rewritten by the authors. The machine-checked Lean formalization of the proof can be found at the GitHub repository (https://github.com/WeihanZhang2001/cubic-root-gaussian-approximation-under-unrestricted-covariance).
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