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The ellipsoid fitting conjecture of Saunderson, Chandrasekaran, Parrilo andWillsky considers the maximum number $n$ random Gaussian points in$\mathbb{R}^d$, such that with high probability, there exists anorigin-symmetric ellipsoid passing through all the points. They conjectured athreshold of $n = (1-o_d(1)) \cdot d^2/4$, while until recently, known lowerbounds on the maximum possible $n$ were of the form $d^2/(\log d)^{O(1)}$. Wegive a simple proof based on concentration of sample covariance matrices, thatwith 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 tworecent independent works by Hsieh, Kothari, Potechin and Xu [arXiv, July 2023]and by Bandeira, Maillard, Mendelson, and Paquette [arXiv, July 2023].

Ellipsoid fitting up to constant via empirical covariance estimation