English(EN) The sharp SAT/UNSAT phase transition in random ellipsoid fitting

数学研究人员证明随机椭球拟合中的尖锐相变

作者 PulseAugur 编辑部 · [1 个来源] · 2026-08-12 04:00

研究人员证明了关于随机椭球拟合中相变的猜想。研究表明，随着维度和数据点数量的增加，在约n ~ d^2/4处发生尖锐相变。在此阈值以下，即使存在特定的谱约束，椭球拟合也可能存在；而在阈值以上，无论谱约束如何，都不可能存在拟合。该证明利用了高斯等价框架，并引入了精确拟合和约束移除的新技术。 AI

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arXiv stat.ML TIER_1 English(EN) · Theodor Misiakiewicz, Garrett G. Wen · 2026-08-12 04:00

随机椭球拟合中的尖锐SAT/UNSAT相变

arXiv:2608.10184v1 Announce Type: cross Abstract: 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 cent…