A Nearly Tight Bound for Fitting an Ellipsoid to Gaussian Random Points
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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 ℝ^d lie on a common ellipsoid with high probability. This nearly establishes a conjecture of <cit.>, 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.
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