Near-optimal fitting of ellipsoids to random points

• URL: http://arxiv.org/abs/2208.09493v4
• Date: Thu, 1 Jun 2023 16:16:33 GMT
• Title: Near-optimal fitting of ellipsoids to random points
• Authors: Aaron Potechin, Paxton Turner, Prayaag Venkat, Alexander S. Wein
• Abstract summary: A basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some $n = Omega(, d2/mathrmpolylog(d),)$. Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a convenient decomposition of a certain non-standard random matrix.
• Score: 68.12685213894112
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Given independent standard Gaussian points $v_1, \ldots, v_n$ 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 $n \sim d^2/4$. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some $n = \Omega( \, d^2/\mathrm{polylog}(d) \,)$, improving prior work of Ghosh et al. [Proc. of Symposium on Foundations of Computer Science, pp. 954-965, 2020] that requires $n = o(d^{3/2})$. 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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