论文标题
唯一性，稀疏性和聚类的几何形状
The Geometry of Uniqueness, Sparsity and Clustering in Penalized Estimation
论文作者
论文摘要
我们为受惩罚的最小二乘估计量的唯一性提供了必要的条件，其惩罚项由标准用多层单位球给出，涵盖了各种方法，包括坡度，PACS，PACS，融合，聚集，经典的和经典的套索以及相关的基础追击方法。我们认为与统计问题相关的强烈唯一性类型。独特条件是几何形状，涉及设计矩阵的行跨度如何与双静态单位球的面相交，而斜率是由签名的固定下给定的。基于此条件的进一步考虑还可以得出稀疏性和聚类特征的结果。特别是，我们定义了斜率模式的概念，以描述该方法的稀疏性和聚类特性，还提供了可访问斜率模式的几何表征。
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.
