Pattern recovery by SLOPE

TL;DR

This paper investigates the theoretical conditions under which the SLOPE method can accurately recover the pattern of regression coefficients, including sparsity and clustering, in high-dimensional regression models.

Contribution

It provides a necessary and sufficient condition for the exact recovery of the SLOPE pattern of regression coefficients, advancing understanding of SLOPE's theoretical properties.

Findings

Derived a necessary and sufficient condition for pattern recovery

Characterized SLOPE's ability to identify sparsity and clustering

Enhanced theoretical understanding of SLOPE in high-dimensional settings

Abstract

SLOPE is a popular method for dimensionality reduction in the high-dimensional regression. Indeed some regression coefficient estimates of SLOPE can be null (sparsity) or can be equal in absolute value (clustering). Consequently, SLOPE may eliminate irrelevant predictors and may identify groups of predictors having the same influence on the vector of responses. The notion of SLOPE pattern allows to derive theoretical properties on sparsity and clustering by SLOPE. Specifically, the SLOPE pattern of a vector provides: the sign of its components (positive, negative or null), the clusters (indices of components equal in absolute value) and clusters ranking. In this article we give a necessary and sufficient condition for SLOPE pattern recovery of an unknown vector of regression coefficients.