Universality and sharp thresholds for ellipsoid fitting

2026-08-27

AI Summary

研究解决了随机向量椭球拟合问题，通过确定一个明确的可满足阈值，使用正定椭球拟合方法，并揭示了四阶矩普遍性现象。

Abstract

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.

Problem

• ellipsoid fitting
• phase transition
• random vectors
• satisfiability threshold
• fourth moment

Innovation

• sharp phase transition
• ellipsoid fitting
• fourth moment universality
• satisfiability threshold
• subgaussian coordinates