Link between graduated-quantile screening and high-dimensional regularity conditions

Establish a quantitative link between the graduated-quantile screening schedule used by Sequential Supersaturated Screening and standard high-dimensional regularity conditions, specifically the irrepresentable condition and restricted eigenvalue conditions.

Background

Sequential Supersaturated Screening provides a conditional bound on the probability that an inactive factor is committed to the selected set. The bound depends on assumptions controlling inactive-factor survival through elimination rounds and commitment by the elbow rule.

The paper does not derive conditions connecting the graduated-quantile schedule to established high-dimensional conditions used in sparse-regression theory, such as the irrepresentable condition and restricted eigenvalue conditions. Establishing such a connection would strengthen the theoretical justification of the screening procedure beyond its current conditional guarantees.