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Near-optimal fitting of ellipsoids to random points

Abstract

Given independent standard Gaussian points v1,,vnv_1, \ldots, v_n in dimension dd, for what values of (n,d)(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 nn increases, with a sharp threshold at nd2/4n \sim d^2/4. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some n=Ω(d2/polylog(d))n = \Omega( \, d^2/\mathrm{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)n = o(d^{3/2}). 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.

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