In this paper, we study properties of penalized and structured M-estimators of multivariate scatter, based on geodesically convex but not necessarily smooth penalty functions. Existence and uniqueness conditions for these penalized and structured estimators are given. However, we show that the standard fixed-point algorithm which is usually applied to an M-estimation problem does not necessarily converge for penalized M-estimation problems. Hence, we develop a new but simple re-weighting algorithm and prove that it has monotone convergence for a broad class of penalized and structured M-estimators of multivariate scatter.

翻译：本文研究了基于测地凸但不一定光滑的罚函数的多元散布的惩罚性和结构性M估计量的性质。给出了这些惩罚性和结构性估计量存在且唯一存在的条件。然而，我们证明了通常应用于M估计问题的标准不动点算法对于惩罚性M估计问题不一定收敛。因此，我们开发了一种新的简单重加权算法，并证明该算法对于一大类多元散布的惩罚性和结构性M估计量具有单调收敛性。