We prove that for $c>0$ a sufficiently small universal constant that a random set 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 a conjecture of~\cite{ SaundersonCPW12}, 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.
Categories: math.PR cs.DS cs.LG math.ST stat.ML stat.TH