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：我们考虑了将中心化椭球拟合到 $\mathbb{R}^d$ 中 $n$ 个标准高斯随机向量的问题，其中 $n, d \to \infty$ 且 $n/d^2 \to \alpha > 0$。已有猜想认为，该问题在高概率下，当 $\alpha < 1/4$ 时可满足（SAT；即存在一个椭球穿过所有 $n$ 个点），而当 $\alpha > 1/4$ 时不可满足（UNSAT）。本文基于统计物理学中非严格的副本方法，给出了一个精确的分析论证，预测了在 $\alpha = 1/4$ 处确实存在 SAT/UNSAT 转变，并刻画了 SAT 阶段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”的稀释极限。我们进一步研究了描述椭球矩阵的不同显式算法构造。特别地，我们证明了一种基于最小化其核范数的过程可在整个 SAT 阶段得到解。最后，我们刻画了一大类旋转不变随机向量椭球拟合的 SAT/UNSAT 转变。我们的工作为分析随机向量椭球拟合提供了数学上严谨的方法，这也是后续工作的主题。