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

TL;DR

This paper proves that a small set of Gaussian points in high-dimensional space almost surely lie on a common ellipsoid, advancing understanding of geometric properties relevant to machine learning and sum-of-squares problems.

Contribution

It establishes a nearly tight bound on the number of Gaussian points needed to fit an ellipsoid, nearly confirming a longstanding conjecture.

Findings

A set of $c d^2/ ext{log}^4(d)$ Gaussian points lie on a common ellipsoid with high probability.

The result nearly confirms a conjecture related to geometric fitting in high dimensions.

Connections to machine learning and sum-of-squares lower bounds are discussed.

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.