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Fitting an ellipsoid to a quadratic number of random points

Afonso S. Bandeira, Antoine Maillard, Shahar Mendelson, Elliot Paquette

TL;DR

This work improves over previous approaches using a key result of Bartl&Mendelson on the concentration of Gram matrices of random vectors under mild assumptions on their tail behavior, and gives a simple proof that the problem of fitting standard Gaussian random vectors to the boundary of a centered ellipsoid is feasible with high probability.

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 $C > 0$.

Fitting an ellipsoid to a quadratic number of random points

TL;DR

This work improves over previous approaches using a key result of Bartl&Mendelson on the concentration of Gram matrices of random vectors under mild assumptions on their tail behavior, and gives a simple proof that the problem of fitting standard Gaussian random vectors to the boundary of a centered ellipsoid is feasible with high probability.

Abstract

We consider the problem of fitting standard Gaussian random vectors in to the boundary of a centered ellipsoid, as . This problem is conjectured to have a sharp feasibility transition: for any , if then has a solution with high probability, while has no solutions with high probability if . So far, only a trivial bound is known on the negative side, while the best results on the positive side assume . 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 is feasible with high probability when , for a (possibly large) constant .
Paper Structure (18 sections, 15 theorems, 85 equations)

This paper contains 18 sections, 15 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Motivation and related literature
  3. The dual semidefinite program
  4. Proof of Theorem \ref{['thm:main']}
  5. Identity perturbation ansatz
  6. Concentration of a kernel Gram matrix
  7. Reducing to a net
  8. Controlling points on the net
  9. Ending the proof
  10. Auxiliary proofs
  11. Proof of Lemma \ref{['lemma:operator_norm_Theta']}
  12. Proof of Lemma \ref{['lemma:Hanson_Wright']}
  13. Tail bounds for chi2 random variables
  14. Proof of Lemma \ref{['lemma:inverse_op']}
  15. Proof of Lemma \ref{['lemma:high_p_events']}
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $n, d \geq 1$, and $x_1, \cdots, x_n \overset{\mathrm{i.i.d.}}{\sim} \mcN(0, \mathrm{I}_d/d)$. Given any $\beta \geq 1$, there exist a (small) constant $\alpha = \alpha(\beta) > 0$ and a (large) constant $C = C(\beta) > 0$ such that for $n \leq \alpha d^2$:

Theorems & Definitions (18)

  • Conjecture 1.1: The ellipsoid fitting conjecture
  • Theorem 1.2: Ellipsoid fitting up to a constant
  • Corollary 1.3: Dual problem
  • Lemma 2.1: Concentration of a kernel Gram matrix
  • Corollary 2.2: Concentration of the inverse
  • Lemma 2.3
  • Lemma 2.4: Some high-probability events
  • Lemma 2.5: Truncating and centering $\tilde{q}$
  • Lemma 2.6
  • Lemma 2.7
  • ...and 8 more