Fitting an ellipsoid to a quadratic number of random points

• URL: http://arxiv.org/abs/2307.01181v2
• Date: Wed, 02 Oct 2024 15:13:40 GMT
• Title: Fitting an ellipsoid to a quadratic number of random points
• Authors: Afonso S. Bandeira, Antoine Maillard, Shahar Mendelson, Elliot Paquette,
• Abstract summary: We consider the problem $(mathrmP)$ of fitting $n$ standard Gaussian random vectors in $mathbbRd$ 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 $n leq (1 - varepsilon) d2 / 4$ then $(mathrmP)$ has a solution with high probability.
• Score: 10.208117253395342
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• 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 $C > 0$.

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