AI 中文总结
该研究解决了Saunderson等人提出的椭球拟合猜想，证明维度为d的m个独立高斯点的椭球拟合问题在m=d²/4处存在尖锐相变。
AI 中文摘要
我们在一个趋于零的因子范围内解决了Saunderson、Chandrasekaran、Parrilo和Willsky提出的椭球拟合猜想。具体来说，对于维度为d的m个独立高斯点，我们证明：当m ≤ (1-o_d(1))·d²/4时，大概率存在一个中心椭球经过所有m个点；当m ≥ (1+o_d(1))·d²/4时，不存在这样的椭球。这确认了椭球拟合问题在d²/4处存在尖锐相变。
英文摘要
We resolve the ellipsoid fitting conjecture of Saunderson, Chandrasekaran, Parrilo, and Willsky up to a vanishing factor. Concretely, for $m$ independent Gaussian points in dimension $d$, we show that with high probability, for $m \leq (1-o_d(1)) \cdot d^2/4$, there exists a centered ellipsoid passing through all $m$ points; for $m\geq (1+o_d(1) )\cdot d^2/4$, no such ellipsoid exists. This confirms that the ellipsoid fitting problem has a sharp phase transition at $d^2/4$.
