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Given independent standard Gaussian points v1, . . . , 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 (Saunderson, 2011; Saunderson et al., 2013) conjectured 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 = d^2/polylog(d). Our proof demonstrates feasibility of the least squares construction of (Saunderson, 2011; Saunderson et al., 2013) 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.