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 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：我们考虑问题(P)：将n个标准高斯随机向量（在R^d中）拟合到一个中心椭球的边界上，其中n, d → ∞。该问题被猜想存在一个尖锐的可行性转变：对于任意ε > 0，若n ≤ (1 - ε)d^2 / 4，则(P)以高概率有解；若n ≥ (1 + ε)d^2 /4，则(P)以高概率无解。迄今为止，在否定一侧仅已知平凡界n ≥ d^2 / 2，而在肯定一侧的最佳结果假设n ≤ d^2 / polylog(d)。本文利用Bartl和Mendelson关于随机向量Gram矩阵在尾部行为温和假设下浓度性质的关键结果，改进了先前方法。我们由此给出一个简洁证明：当n ≤ d^2 / C（其中C > 0为可能较大的常数）时，(P)以高概率可行。