Ellipsoid fitting up to constant via empirical covariance estimation

TL;DR

This paper proves that with high probability, a constant-factor number of random Gaussian points in high dimensions can be enclosed by an ellipsoid, advancing the understanding of the ellipsoid fitting conjecture.

Contribution

The authors provide a simple proof using empirical covariance estimation to establish a new lower bound on the number of Gaussian points that can be fitted by an ellipsoid.

Findings

Fitted ellipsoid through approximately d^2 points with high probability

Improved lower bounds on ellipsoid fitting thresholds

Results align with recent independent works

Abstract

The ellipsoid fitting conjecture of Saunderson, Chandrasekaran, Parrilo and Willsky considers the maximum number $n$ random Gaussian points in $\mathbb{R}^d$, such that with high probability, there exists an origin-symmetric ellipsoid passing through all the points. They conjectured a threshold of $n = (1-o_d(1)) \cdot d^2/4$, while until recently, known lower bounds on the maximum possible $n$ were of the form $d^2/(\log d)^{O(1)}$. We give a simple proof based on concentration of sample covariance matrices, that with probability $1 - o_d(1)$, it is possible to fit an ellipsoid through $d^2/C$ random Gaussian points. Similar results were also obtained in two recent independent works by Hsieh, Kothari, Potechin and Xu [arXiv, July 2023] and by Bandeira, Maillard, Mendelson, and Paquette [arXiv, July 2023].