AI 中文总结
本研究证明了随机椭球拟合的半定可行性问题在$n \sim d^2/4$处存在尖锐SAT/UNSAT相变，基于高斯等价框架完成证明，填补了相关工作的空白。
AI 中文摘要
设$x_1,\ldots,x_n$为$\mathbb{R}^d$中的独立标准高斯向量。椭球拟合是满足$S \succeq 0$的矩阵$S$，使得对每个$i$有$x_i^\top S x_i =d$，即所有点都位于中心椭球$\{ x: x^\top S x = d\}$的边界上。Saunderson、Parrilo和Willsky推测，当$n,d \to \infty$时，该半定可行性问题在$n \sim d^2/4$处发生尖锐相变。我们证明了这一猜想：若$\lim \sup n/d^2 = \alpha^* <1/4$，则概率趋于1时存在椭球拟合；此外，可选择所有特征值位于仅依赖$\alpha^*$的固定区间$[\lambda_-, \lambda_+] \subset (0,\infty)$内的$S$。反之，若$\lim \inf n/d^2 > 1/4$，则概率趋于1时不存在任何椭球拟合，且无谱限制。我们的证明基于Bandeira和Maillard（2025）提出的高斯等价框架，填补了该工作留下的两个空白：建立精确拟合并移除算子范数约束。在可满足侧，新的要素包括对偶向量的头尾分解、稀疏头约束的精确修正，以及低影响尾的高斯比较原理。在不可满足侧，我们将候选解拆分为低秩谱头和Schatten-3弥散体，在头的条件下对体进行高斯化，并应用投影Gordon逃逸论证。该阈值由半正定锥的统计维度$d(d+1)/4$决定。
英文摘要
Let $x_1,\ldots,x_n$ be independent standard Gaussian vectors in $\mathbb{R}^d$. An \emph{ellipsoid fit} is a matrix $S \succeq 0$ such that $x_i^\top S x_i =d$ for every $i$, so that all the points lie on the boundary of the centered ellipsoid $\{ x : x^\top S x = d\}$. Saunderson, Parrilo and Willsky conjectured that, as $n,d \to \infty$, this semidefinite feasibility problem undergoes a sharp transition at $n \sim d^2/4$. We prove this conjecture. If $\lim \sup n/d^2 = α^* <1/4$, then, with probability tending to one, an ellipsoid fit exists; moreover, one can choose $S$ with all eigenvalues in a fixed interval $[λ_- , λ_+] \subset (0,\infty)$ depending only on $α^*$. Conversely, if $\lim \inf n/d^2 > 1/4$, then, with probability tending to one, no ellipsoid fit exists, without any spectral restriction. Our proof builds on the Gaussian-equivalence framework developed by Bandeira and Maillard (2025) and closes the two gaps left open in their work: establishing exact fitting and removing the operator-norm constraint. On the satisfiable side, the new ingredients are a head-tail decomposition of the dual vector, exact correction of the sparse head constraints, and a Gaussian comparison principle for the low-influence tail. On the unsatisfiable side, we split a candidate into a low-rank spectral head and a Schatten-3 diffuse bulk, Gaussianize the bulk conditionally on the head, and apply a projected Gordon escape argument. The threshold is governed by the statistical dimension $d(d+1)/4$ of the positive semidefinite cone.
