We consider the problem of fitting a centered ellipsoid to $n$ standard Gaussian random vectors in $\mathbb{R}^d$, as $n, d \to \infty$ with $n/d^2 \to \alpha > 0$. It has been conjectured that this problem is, with high probability, satisfiable (SAT; that is, there exists an ellipsoid passing through all $n$ points) for $\alpha < 1/4$, and unsatisfiable (UNSAT) for $\alpha > 1/4$. In this work we give a precise analytical argument, based on the non-rigorous replica method of statistical physics, that indeed predicts a SAT/UNSAT transition at $\alpha = 1/4$, as well as the shape of a typical fitting ellipsoid in the SAT phase (i.e., the lengths of its principal axes). Besides the replica method, our main tool is the dilute limit of extensive-rank "HCIZ integrals" of random matrix theory. We further study different explicit algorithmic constructions of the matrix characterizing the ellipsoid. In particular, we show that a procedure based on minimizing its nuclear norm yields a solution in the whole SAT phase. Finally, we characterize the SAT/UNSAT transition for ellipsoid fitting of a large class of rotationally-invariant random vectors. Our work suggests mathematically rigorous ways to analyze fitting ellipsoids to random vectors, which is the topic of a companion work.

翻译：我们考虑将中心化椭球拟合至$\mathbb{R}^d$中$n$个标准高斯随机向量的问题，其中$n, d \to \infty$且$n/d^2 \to \alpha > 0$。此前有猜想认为：当$\alpha < 1/4$时，该问题高概率可满足（SAT，即存在椭球通过所有$n$个点）；当$\alpha > 1/4$时则不可满足（UNSAT）。本文基于统计物理学中非严格的副本方法，给出精确解析论证，不仅预测了$\alpha = 1/4$处的SAT/UNSAT相变，还刻画了SAT相中典型拟合椭球的形状（即其主轴长度）。除副本方法外，我们的主要工具是随机矩阵理论中广秩"HCIZ积分"的稀薄极限。我们进一步研究了表征椭球矩阵的不同显式算法构造，特别证明了基于核范数最小化的过程能在整个SAT相中得到解。最后，我们刻画了旋转不变随机向量大类椭球拟合问题的SAT/UNSAT相变。本工作为分析随机向量椭球拟合提供了数学上可严格化的途径，这也是后续研究的主题。