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].

翻译：桑德森、钱德拉塞卡兰、帕里洛和威尔斯基提出的椭球拟合猜想，探讨了在$\mathbb{R}^d$空间中随机高斯点$n$的最大数量，使得存在一个过原点的对称椭球以高概率穿过所有点。他们推测阈值为$n = (1-o_d(1)) \cdot d^2/4$，而直到最近，已知$n$最大可能值的下界仍为$d^2/(\log d)^{O(1)}$的形式。我们基于样本协方差矩阵的集中性给出了一个简单证明：以概率$1 - o_d(1)$，能够拟合一个椭球穿过$d^2/C$个随机高斯点。类似结果也在近期两项独立研究——谢西恩、科塔里、波泰钦和徐 [arXiv, 2023年7月] 以及班代拉、马亚尔、门德尔松和帕凯特 [arXiv, 2023年7月] 中获得。