发表机构
University of Chicago; Brown University（芝加哥大学; 布朗大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对随机向量的椭球拟合问题，确定了依赖四阶矩的显式可满足阈值，标准高斯数据下阈值为1/4，解决了椭球拟合猜想并揭示了普适性现象。
AI 中文摘要
我们建立了用椭球拟合随机向量的尖锐相变。这些随机向量具有独立的次高斯坐标，均值为零、方差为1，且具有共同的四阶矩，向量数量与维度的平方成正比。我们确定了一个显式可满足阈值：在阈值以下，大概率存在一个正定椭球经过所有数据点；而在阈值以上，不存在正半定拟合。我们还确定了不可满足区域内的最优平方拟合误差。特别地，该阈值仅通过共同四阶矩依赖于坐标分布，揭示了四阶矩普适性现象。对于标准高斯数据，该阈值为1/4，解决了椭球拟合猜想。
英文摘要
We establish a sharp phase transition for fitting random vectors by an ellipsoid. The random vectors have independent subgaussian coordinates with mean zero, variance one, and a common fourth moment, and the number of vectors is proportional to the square of the dimension. We identify an explicit satisfiability threshold such that, with high probability, a positive definite ellipsoid passes through every data point below the threshold, whereas no positive semidefinite fit exists above it. We also determine the optimal squared fitting error throughout the unsatisfiable regime. In particular, the threshold depends on the coordinate distributions only through their common fourth moment, revealing a fourth moment universality phenomenon. For standard Gaussian data the threshold is $1/4$, resolving the ellipsoid fitting conjecture.
Comments79 pages, 4 figures
