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This paper investigates the limitations of restricted eigenvalue (RE) bounds in the context of heavy-tailed designs, revealing that the previously established sample size benchmark does not hold under these conditions. The authors demonstrate that simultaneous threshold occupancy is a critical obstruction, leading to a stark contrast in sample complexity between Gaussian measurements and isotropic heavy-tailed designs. Specifically, while Gaussian measurements require only logarithmic samples, heavy-tailed designs necessitate significantly more, highlighting the nuanced challenges in statistical recovery under non-Gaussian conditions.
Heavy-tailed designs can lead to a dramatic increase in sample complexity, challenging established norms in statistical recovery.
Restricted eigenvalue (RE) bounds govern stable recovery by norm-regularized estimators. For isotropic sub-Gaussian measurements, the benchmark sample size is $1+w(A)^2$, where $w(A)$ is the Gaussian width of the normalized descent cone. The COLT 2015 open-problem note (Banerjee et al., 2015) asked whether the same law follows for heavy-tailed designs from a uniform small-ball condition alone. We give an explicit and systematic negative answer to the general question as formulated there: the proposed law fails in its full dimension-free, arbitrary-set form, and the missing obstruction is simultaneous threshold occupancy. A constant-width polyhedral descent cone with fixed small-ball constants has zero empirical RE on every sample path up to half the ambient dimension. More generally, every finite range space admits exact threshold encoding in an arbitrarily narrow spherical cap and a lift to a full polyhedral descent-cone section. For every fixed threshold VC dimension $d$, as $\beta\downarrow0$, the sharp worst-case sample complexity is $\Theta(\beta^{-1}[d\log(1/\beta)+\log(1/\delta)])$. The separation persists under exact isotropy and all finite moments: on the same constant-width cone, Gaussian measurements succeed with $O(1+\log(1/\delta))$ samples, whereas an isotropic heavy-tailed design fails pathwise for $n\lesssim\sqrt{p/\log p}$. Gaussian smoothing yields an everywhere-positive $C^\infty$ density while retaining arbitrarily poor RE. Under isotropy, a distribution-free fallback governed by affine dimension times squared enclosing radius is sharp on this family.
