Fitting an ellipsoid to a quadratic number of random points

TL;DR

This paper investigates the problem of fitting a centered ellipsoid to a large number of Gaussian points in high dimensions, establishing a near-optimal feasibility threshold using advanced concentration results.

Contribution

It improves the understanding of the feasibility transition for ellipsoid fitting by providing a simpler proof that solutions exist with high probability when the number of points is below a constant times the square of the dimension.

Findings

Feasibility of ellipsoid fitting when n ≤ d^2 / C

Improved bounds on the phase transition for the problem

Application of concentration of Gram matrices to high-dimensional geometry

Abstract

We consider the problem $(\mathrm{P})$ of fitting $n$ standard Gaussian random vectors in $\mathbb{R}^d$ to the boundary of a centered ellipsoid, as $n, d \to \infty$. This problem is conjectured to have a sharp feasibility transition: for any $\varepsilon > 0$, if $n \leq (1 - \varepsilon) d^2 / 4$ then $(\mathrm{P})$ has a solution with high probability, while $(\mathrm{P})$ has no solutions with high probability if $n \geq (1 + \varepsilon) d^2 /4$. So far, only a trivial bound $n \geq d^2 / 2$ is known on the negative side, while the best results on the positive side assume $n \leq d^2 / \mathrm{polylog}(d)$. In this work, we improve over previous approaches using a key result of Bartl & Mendelson (2022) on the concentration of Gram matrices of random vectors under mild assumptions on their tail behavior. This allows us to give a simple proof that $(\mathrm{P})$ is feasible with high probability when $n \leq d^2 / C$, for a (possibly large) constant $C > 0$.