Fitting an ellipsoid to a quadratic number of random points

07/03/2023

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by   Afonso S. Bandeira, et al.

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We consider the problem (P) of fitting n standard Gaussian random vectors in ℝ^d to the boundary of a centered ellipsoid, as n, d →∞. This problem is conjectured to have a sharp feasibility transition: for any ε > 0, if n ≤ (1 - ε) d^2 / 4 then (P) has a solution with high probability, while (P) has no solutions with high probability if n ≥ (1 + ε) d^2 /4. So far, only a trivial bound n ≥ d^2 / 2 is known on the negative side, while the best results on the positive side assume n ≤ d^2 / polylog(d). In this work, we improve over previous approaches using a key result of Bartl Mendelson on the concentration of Gram matrices of random vectors under mild assumptions on their tail behavior. This allows us to give a simple proof that (P) is feasible with high probability when n ≤ d^2 / C, for a (possibly large) constant C > 0.