Near-optimal fitting of ellipsoids to random points
share
Given independent standard Gaussian points v_1, …, v_n in dimension d, for what values of (n, d) does there exist with high probability an origin-symmetric ellipsoid that simultaneously passes through all of the points? This basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. Based on strong numerical evidence, Saunderson, Parrilo, and Willsky [Proc. of Conference on Decision and Control, pp. 6031-6036, 2013] conjecture that the ellipsoid fitting problem transitions from feasible to infeasible as the number of points n increases, with a sharp threshold at n ∼ d^2/4. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some n = Ω( d^2/log^5(d) ), improving prior work of Ghosh et al. [Proc. of Symposium on Foundations of Computer Science, pp. 954-965, 2020] that requires n = o(d^3/2). Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a careful analysis of the eigenvectors and eigenvalues of a certain non-standard random matrix.
READ FULL TEXT