Restricted-eigenvalue bounds under isotropic non-sub-Gaussian designs

Determine whether a small-ball condition, together with isotropy of the random design, suffices to establish the proposed Euclidean Gaussian-width restricted-eigenvalue bound for spherical restricted sets, or otherwise characterize the additional assumptions required.

Background

The paper disproves a distribution-free formulation in which a marginal small-ball lower bound alone yields restricted-eigenvalue sample complexity governed by the ordinary Euclidean Gaussian width. Its counterexample uses a centered, finite-covariance, heavy-tailed, anisotropic row distribution, so it does not address the stronger setting in which the design is isotropic.

The authors explicitly state that the proof does not settle a formulation assuming isotropy from the outset. They also explain that simply whitening the construction changes the geometry of the restricted directions and destroys the bounded-width estimate, so the isotropic case requires separate analysis.