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Random sections of $\ell_p$-ellipsoids, optimal recovery and Gelfand numbers of diagonal operators

Aicke Hinrichs, Joscha Prochno, Mathias Sonnleitner

TL;DR

An upper bound for random sections is proved using techniques from asymptotic geometric analysis if 1 ≤ p ≤ ∞ and compressed sensing if 0 < p ≤ 1, which can be interpreted as a bound on the quality of random (Gaussian) information for the recovery of vectors from an lp-ellipsoid for which the radius of optimal information is given by the Gelfand numbers of a diagonal operator.

Abstract

We study the circumradius of a random section of an $\ell_p$-ellipsoid, $0<p\le \infty$, and compare it with the minimal circumradius over all sections with subspaces of the same codimension. Our main result is an upper bound for random sections, which we prove using techniques from asymptotic geometric analysis if $1\leq p \leq \infty$ and compressed sensing if $0<p \leq 1$. This can be interpreted as a bound on the quality of random (Gaussian) information for the recovery of vectors from an $\ell_p$-ellipsoid for which the radius of optimal information is given by the Gelfand numbers of a diagonal operator. In the case where the semiaxes decay polynomially and $1\le p\le \infty$, we conjecture that, as the amount of information increases, the radius of random information either decays like the radius of optimal information or is bounded from below by a constant, depending on whether the exponent of decay is larger than the critical value $1-\frac{1}{p}$ or not. If $1\leq p\leq 2$, we prove this conjecture by providing a matching lower bound. This extends the recent work of Hinrichs et al. [Random sections of ellipsoids and the power of random information, Trans. Amer. Math. Soc., 2021+] for the case $p=2$.

Random sections of $\ell_p$-ellipsoids, optimal recovery and Gelfand numbers of diagonal operators

TL;DR

An upper bound for random sections is proved using techniques from asymptotic geometric analysis if 1 ≤ p ≤ ∞ and compressed sensing if 0 < p ≤ 1, which can be interpreted as a bound on the quality of random (Gaussian) information for the recovery of vectors from an lp-ellipsoid for which the radius of optimal information is given by the Gelfand numbers of a diagonal operator.

Abstract

We study the circumradius of a random section of an -ellipsoid, , and compare it with the minimal circumradius over all sections with subspaces of the same codimension. Our main result is an upper bound for random sections, which we prove using techniques from asymptotic geometric analysis if and compressed sensing if . This can be interpreted as a bound on the quality of random (Gaussian) information for the recovery of vectors from an -ellipsoid for which the radius of optimal information is given by the Gelfand numbers of a diagonal operator. In the case where the semiaxes decay polynomially and , we conjecture that, as the amount of information increases, the radius of random information either decays like the radius of optimal information or is bounded from below by a constant, depending on whether the exponent of decay is larger than the critical value or not. If , we prove this conjecture by providing a matching lower bound. This extends the recent work of Hinrichs et al. [Random sections of ellipsoids and the power of random information, Trans. Amer. Math. Soc., 2021+] for the case .

Paper Structure

This paper contains 11 sections, 19 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction, motivation and main results
  2. Radii of random sections and optimal recovery
  3. Radii of random sections -- main results
  4. The power of random information for polynomial decay -- discussion
  5. An upper bound via an $M^{*}$-estimate -- the case $1\leq p\leq \infty$
  6. An $M^*$-estimate for $\mathcal{E}_{p,\sigma}^{m}\cap \varrho \mathbb{B}_{2}^{m}$
  7. The proof of Theorem \ref{['thm:mstar-estimate-rad']}
  8. An upper bound via compressed sensing techniques -- the case $0<p<1$
  9. Elements from compressed sensing and bounds on the best $s$-term approximation
  10. The proof of Theorem \ref{['thm:radius-quasi']}
  11. A lower bound -- the case $1< p\le 2$

Key Result

Theorem A

For all $m\in\mathbb{N}$ and $1\le n<m$, we have with probability at least $1-c_1 \exp(-c_2 n)$, where $c_1,c_2\in(0,\infty)$ are absolute constants, and $k\asymp n$ for $p=1$ while $k\asymp \frac{n}{p^*}$ for $p>1$.

Figures (1)

  • Figure :

Theorems & Definitions (34)

  • Theorem A
  • Remark 1
  • Remark 2
  • Theorem B
  • Proposition 1
  • Remark 3
  • Corollary 1
  • Conjecture
  • Proposition 2
  • Lemma 1
  • ...and 24 more