# Universality and sharp thresholds for ellipsoid fitting

arXiv: https://arxiv.org/abs/2608.27372 · HTML: https://mlsanity.com/paper/2608.27372/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 82 references that are in the map.
- First posted on arXiv 27 Aug 2026.

## Builds on (most cited first, 3 of 3)

- [Universality of empirical risk minimization](https://mlsanity.com/paper/2202.08832/) (2022)
- [Universality of max-margin classifiers](https://mlsanity.com/paper/2310.00176/) (2023)
- [The sharp SAT/UNSAT phase transition in random ellipsoid fitting](https://mlsanity.com/paper/2608.10184/) (2026)

## Abstract (arXiv)

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.