We consider the problem $(\mathrm{P})$ of fitting $n$ standard Gaussian random vectors in $\mathbb{R}^d$ to the boundary of a centered ellipsoid, as $n, d \to \infty$. This problem is conjectured to have a sharp feasibility transition: for any $\varepsilon > 0$, if $n \leq (1 - \varepsilon) d^2 / 4$ then $(\mathrm{P})$ has a solution with high probability, while $(\mathrm{P})$ has no solutions with high probability if $n \geq (1 + \varepsilon) d^2 /4$. So far, only a trivial bound $n \geq d^2 / 2$ is known on the negative side, while the best results on the positive side assume $n \leq d^2 / \mathrm{polylog}(d)$. In this work, we improve over previous approaches using a key result of Bartl & Mendelson (2022) on the concentration of Gram matrices of random vectors under mild assumptions on their tail behavior. This allows us to give a simple proof that $(\mathrm{P})$ is feasible with high probability when $n \leq d^2 / C$, for a (possibly large) constant $C > 0$.

翻译：我们考虑将 $n$ 个 $\mathbb{R}^d$ 中的标准高斯随机向量拟合到一个中心椭球边界的问题 $(\mathrm{P})$，其中 $n, d \to \infty$。该问题被推测存在一个尖锐的可行性相变：对于任意 $\varepsilon > 0$，若 $n \leq (1 - \varepsilon) d^2 / 4$，则 $(\mathrm{P})$ 以高概率存在解；而若 $n \geq (1 + \varepsilon) d^2 /4$，则 $(\mathrm{P})$ 以高概率无解。迄今为止，负侧仅知平凡下界 $n \geq d^2 / 2$，而正侧的最佳结果需假设 $n \leq d^2 / \mathrm{polylog}(d)$。本工作中，我们利用 Bartl & Mendelson (2022) 关于随机向量 Gram 矩阵在尾部行为温和假设下浓度的一个关键结果，改进了先前方法。这使我们能够给出一个简洁证明：当 $n \leq d^2 / C$（其中 $C > 0$ 为（可能较大的）常数）时，$(\mathrm{P})$ 以高概率可行。