A Nearly Tight Bound for Fitting an Ellipsoid to Gaussian Random Points

• URL: http://arxiv.org/abs/2212.11221v1
• Date: Wed, 21 Dec 2022 17:48:01 GMT
• Title: A Nearly Tight Bound for Fitting an Ellipsoid to Gaussian Random Points
• Authors: Daniel M. Kane and Ilias Diakonikolas
• Abstract summary: This nearly establishes a conjecture ofciteSaundersonCPW12, within logarithmic factors. The latter conjecture has attracted significant attention over the past decade, due to its connections to machine learning and sum-of-squares lower bounds for certain statistical problems.
• Score: 50.90125395570797
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We prove that for $c>0$ a sufficiently small universal constant that a random set of $c d^2/\log^4(d)$ independent Gaussian random points in $\mathbb{R}^d$ lie on a common ellipsoid with high probability. This nearly establishes a conjecture of~\cite{SaundersonCPW12}, within logarithmic factors. The latter conjecture has attracted significant attention over the past decade, due to its connections to machine learning and sum-of-squares lower bounds for certain statistical problems.

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