We consider the problem $(\rm P)$ of exactly fitting an ellipsoid (centered at $0$) to $n$ standard Gaussian random vectors in $\mathbb{R}^d$, as $n, d \to \infty$ with $n / d^2 \to \alpha > 0$. This problem is conjectured to undergo a sharp transition: with high probability, $(\rm P)$ has a solution if $\alpha < 1/4$, while $(\rm P)$ has no solutions if $\alpha > 1/4$. So far, only a trivial bound $\alpha > 1/2$ is known to imply the absence of solutions, while the sharpest results on the positive side assume $\alpha \leq \eta$ (for $\eta > 0$ a small constant) to prove that $(\rm P)$ is solvable. In this work we study universality between this problem and a so-called "Gaussian equivalent", for which the same transition can be rigorously analyzed. Our main results are twofold. On the positive side, we prove that if $\alpha < 1/4$, there exist an ellipsoid fitting all the points up to a small error, and that the lengths of its principal axes are bounded above and below. On the other hand, for $\alpha > 1/4$, we show that achieving small fitting error is not possible if the length of the ellipsoid's shortest axis does not approach $0$ as $d \to \infty$ (and in particular there does not exist any ellipsoid fit whose shortest axis length is bounded away from $0$ as $d \to \infty$). To the best of our knowledge, our work is the first rigorous result characterizing the expected phase transition in ellipsoid fitting at $\alpha = 1/4$. In a companion non-rigorous work, the first author and D. Kunisky give a general analysis of ellipsoid fitting using the replica method of statistical physics, which inspired the present work.

翻译：我们研究从$\mathbb{R}^d$中$n$个标准高斯随机向量精确拟合中心位于原点的椭球的问题$(\rm P)$，其中$n, d \to \infty$且$n / d^2 \to \alpha > 0$。该问题被推测会发生尖锐相变：当$\alpha < 1/4$时，$(\rm P)$高概率存在解；而当$\alpha > 1/4$时，$(\rm P)$无解。目前，仅已知平凡界$\alpha > 1/2$蕴含无解，而正面最精确的结果假设$\alpha \leq \eta$（$\eta > 0$为小常数）才能证明$(\rm P)$可解。本文研究该问题与所谓“高斯等价”问题之间的普适性，后者可严格分析相同的相变。我们的主要结果有两方面。正面结果：我们证明当$\alpha < 1/4$时，存在一个椭球拟合所有点且误差很小，其主轴长度上下有界。另一方面，当$\alpha > 1/4$时，若椭球最短轴长度不随$d \to \infty$趋近于$0$（特别地，不存在任何拟合椭球使其最短轴长度在$d \to \infty$时保持远离$0$），则无法实现小误差拟合。据我们所知，本文首次严格刻画了椭球拟合在$\alpha = 1/4$处的预期相变。在配套的非严格工作中，第一作者与D. Kunisky利用统计物理的复制法对椭球拟合进行了普适分析，这启发了本文的工作。