Ellipsoid fitting up to constant via empirical covariance estimation

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 $n=(1-o_d(1))\cdot d^2\mathrm{/}4$, while until recently, known lower bounds on the maximum possible $n$ were of the form $d^2\mathrm{/}(\log d{)}^{O(1)}$. 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 $d^2\mathrm{/}C$ 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].