Pattern recovery by SLOPE

SLOPE is a popular method for dimensionality reduction in high-dimensional regression. Its estimated coefficients can be zero, yielding sparsity, or equal in absolute value, yielding clustering. As a result, SLOPE can eliminate irrelevant predictors and identify groups of predictors that have the same influence on the response. The concept of the SLOPE pattern allows us to formalize and study its sparsity and clustering properties. In particular, the SLOPE pattern of a coefficient vector captures the signs of its components (positive, negative, or zero), the clusters (groups of coefficients with the same absolute value), and the ranking of those clusters. This is the first paper to thoroughly investigate the consistency of the SLOPE pattern. We establish necessary and sufficient conditions for SLOPE pattern recovery, which in turn enable the derivation of an irrepresentability condition for SLOPE given a fixed design matrix $X$. These results lay the groundwork for a comprehensive asymptotic analysis of SLOPE pattern consistency.