We provide a necessary and sufficient condition for the uniqueness of penalized least-squares estimators whose penalty term is given by a norm with a polytope unit ball, covering a wide range of methods including SLOPE, PACS, fused, clustered and classical LASSO as well as the related method of basis pursuit. We consider a strong type of uniqueness that is relevant for statistical problems. The uniqueness condition is geometric and involves how the row span of the design matrix intersects the faces of the dual norm unit ball, which for SLOPE is given by the signed permutahedron. Further considerations based this condition also allow to derive results on sparsity and clustering features. In particular, we define the notion of a SLOPE pattern to describe both sparsity and clustering properties of this method and also provide a geometric characterization of accessible SLOPE patterns.

翻译：我们为受惩罚的最低限度估计者规定了一个必要和充分的条件,其惩罚期限由具有多管单元球的规范给予,涵盖多种方法,包括SLOPE、PACS、引信、集束和古典LASSO,以及相关的基础追踪方法。我们认为,与统计问题相关的一种很强的独特性是几何性的,涉及设计矩阵的行长跨度如何将双规范单球的面部交叉开来,而对于双规范单管球,经签署的SLOPE,则由签名的顶层提供。基于这一条件的进一步考虑还有助于得出关于宽度和聚集特征的结果。特别是,我们界定了SLOPE模式的概念,以描述这种方法的宽度和聚集特性,并对可获取的SLOPE模式进行几何描述。