Near-optimal fitting of ellipsoids to random points

Given independent standard Gaussian points v1, ldots, vn 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 ~ d2/4. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some n = Omega( , d2/polylog(d) ,), improving prior work of Ghosh et al. [Proc. of Symposium on Foundations of Computer Science, pp. 954-965, 2020] that requires n = o(d3/2). Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a convenient decomposition of a certain non-standard random matrix and a careful analysis of its Neumann expansion via the theory of graph matrices.