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We prove that for $c>0$ a sufficiently small universal constant that a randomset of $c d^2/\log^4(d)$ independent Gaussian random points in $\mathbb{R}^d$lie on a common ellipsoid with high probability. This nearly establishes aconjecture of~\cite{SaundersonCPW12}, within logarithmic factors. The latterconjecture has attracted significant attention over the past decade, due to itsconnections to machine learning and sum-of-squares lower bounds for certainstatistical problems.

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