揭示随机向量椭球拟合的精确临界点与普适性规律
Universality and sharp thresholds for ellipsoid fitting
- 基于独立亚高斯坐标数据，建立拟合可行性阈值
- 当样本量为维度平方时，阈值由四阶矩决定，最优误差可计算
- 适用于高维统计、几何拟合，尤其关注分布鲁棒性研究者
我们建立了随机向量椭球拟合的精确相变现象。数据向量具有均值为零、方差为一、共同四阶矩的独立亚高斯分量，样本数量与维度平方成正比。我们确定了一个显式的可满足阈值：低于该阈值时，几乎必然存在一个正定椭球通过所有数据点；高于该阈值时，不存在半正定拟合。在不可满足区域，我们还求出了最优平方拟合误差。特别地，该阈值仅依赖于各坐标分布的共同四阶矩，揭示了四阶矩的普适性现象。对于标准高斯数据，阈值为1/4，解决了椭球拟合猜想。
原文摘要 · Abstract (English)
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.
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