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This research establishes a sharp phase transition for fitting random vectors with an ellipsoid, proving that satisfiability depends solely on the common fourth moment of the data distribution. By resolving the long-standing conjecture for standard Gaussian data with a threshold of 1/4, the study highlights a universality phenomenon relevant to high-dimensional statistical learning and disordered systems.
Universality and sharp thresholds for ellipsoid fitting
arXiv:2608.27372v1 Announce Type: cross 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.
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.
Probability (math.PR); Disordered Systems and Neural Networks (cond-mat.dis-nn); Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG); Statistics Theory (math.ST)
arXiv:2608.27372 [math.PR]
(or arXiv:2608.27372v1 [math.PR] for this version)
https://doi.org/10.48550/arXiv.2608.27372 arXiv-issued DOI via DataCite (pending registration)
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