Universality and sharp thresholds for ellipsoid fitting
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, reve...
| Authors | Frederic Koehler, Youngtak Sohn |
|---|---|
| Primary category | math.PR |
| Published | 27 August 2026 |
| Last updated at source | 27 August 2026 |
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