基于经验协方差估计的常数阶椭球拟合
Ellipsoid fitting up to constant via empirical covariance estimation
AI总结：
本研究针对椭球拟合猜想中随机高斯点最大数量的下界问题，基于样本协方差矩阵集中性给出简单证明，将下界提升至$d^2/C$的常数阶，相关结果与同期两项独立工作结论一致。
摘要：
Saunderson、Chandrasekaran、Parrilo与Willsky提出的椭球拟合猜想，关注$\u211d^d$空间中使得高概率存在过所有点的原点对称椭球的随机高斯点的最大数量$n$。他们猜想阈值为$n = (1-o_d(1)) \cdot d^2/4$，但直到近期，已知的最大$n$的下界形式为$d^2/(\log d)^{O(1)}$。我们基于样本协方差矩阵的集中性给出了一个简单证明：以$1 - o_d(1)$的概率，可对$d^2/C$个随机高斯点拟合出一个椭球。类似结果也在Hsieh、Kothari、Potechin与Xu（arXiv，2023年7月）以及Bandeira、Maillard、Mendelson与Paquette（arXiv，2023年7月）的两项近期独立工作中被得到。
英文摘要：
The ellipsoid fitting conjecture of Saunderson, Chandrasekaran, Parrilo and Willsky considers the maximum number $n$ random Gaussian points in $\mathbb{R}^d$, such that with high probability, there exists an origin-symmetric ellipsoid passing through all the points. They conjectured a threshold of $n = (1-o_d(1)) \cdot d^2/4$, while until recently, known lower bounds on the maximum possible $n$ were of the form $d^2/(\log d)^{O(1)}$. We give a simple proof based on concentration of sample covariance matrices, that with probability $1 - o_d(1)$, it is possible to fit an ellipsoid through $d^2/C$ random Gaussian points. Similar results were also obtained in two recent independent works by Hsieh, Kothari, Potechin and Xu [arXiv, July 2023] and by Bandeira, Maillard, Mendelson, and Paquette [arXiv, July 2023].
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