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.

翻译：SLOPE是一种在高维回归中广泛应用的降维方法。事实上，SLOPE的部分回归系数估计值可以为零（稀疏性），或绝对值相等（聚类特性）。因此，SLOPE能够剔除无关预测变量，并识别出对响应向量具有相同影响效应的预测变量组。SLOPE模式这一概念使得我们可以推导SLOPE在稀疏性与聚类方面的理论性质。具体而言，向量的SLOPE模式包含：各分量的符号（正、负或零）、聚类（绝对值相等的分量索引）以及聚类排序。本文给出了未知回归系数向量SLOPE模式恢复的充分必要条件。