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

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 $p=2$.