# A Nearly Tight Bound for Fitting an Ellipsoid to Gaussian Random Points

arXiv: https://arxiv.org/abs/2212.11221 · HTML: https://mlsanity.com/paper/2212.11221/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 9 references that are in the map.
- First posted on arXiv 21 Dec 2022.

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

- [Overcomplete Independent Component Analysis via SDP](https://mlsanity.com/paper/1901.08334/) (2019)
- [Near-optimal fitting of ellipsoids to random points](https://mlsanity.com/paper/2208.09493/) (2022)
- [Algorithmic Thresholds for Refuting Random Polynomial Systems](https://mlsanity.com/paper/2110.08677/) (2021)

## Built on by (most cited first, 2 of 2)

- [Fitting an ellipsoid to a quadratic number of random points](https://mlsanity.com/paper/2307.01181/) (2023)
- [The sharp SAT/UNSAT phase transition in random ellipsoid fitting](https://mlsanity.com/paper/2608.10184/) (2026)

## Abstract (arXiv)

We prove that for c>0 a sufficiently small universal constant that a random set of c d2/log4(d) independent Gaussian random points in ℝd lie on a common ellipsoid with high probability. This nearly establishes a conjecture of~citeSaundersonCPW12, within logarithmic factors. The latter conjecture has attracted significant attention over the past decade, due to its connections to machine learning and sum-of-squares lower bounds for certain statistical problems.