标题： 一般损失函数下由非可微正则化器诱导的低维模式的渐近分布 显示英文标题

标题： Asymptotic Distribution of Low-Dimensional Patterns Induced by Non-Differentiable Regularizers under General Loss Functions

摘要： 本文研究了旨在恢复低维结构的不可微惩罚估计量的渐近分布；也就是说，这些估计量所在的子空间，其中真实参数存在。 我们研究了在参数维度p固定且观测数量n趋于无穷的框架下，缩放后的估计误差的渐近分布。 我们的重点在于模式恢复的渐近概率，这是LASSO的经典结果未涉及的问题。 在我们最近的工作中，我们推导出了LASSO以及更广泛的惩罚类别的结果，包括标准线性模型中的非可分离惩罚，例如SLOPE。 我们现在将这种分析扩展到一般的损失函数，包括例如使用Huber和分位数损失函数的鲁棒回归，以及指数族中的广义线性模型。 本文的主要贡献是开发了一种渐近框架，用于一般损失函数下的正则M估计器的模式收敛，这些损失函数满足适当的随机可微性条件。 证明依赖于经验过程理论中的工具，包括Donsker类和VC维技术。 显示英文摘要

摘要： This article investigates the asymptotic distribution of penalized estimators with non-differentiable penalties designed to recover low-dimensional structures; that is, subspaces in which the true parameter lies. We study the asymptotic distribution of the scaled estimation error in the regime where the parameter dimension p is fixed and the number of observations n tends to infinity. Our focus is on the asymptotic probability of pattern recovery, a question not addressed by classical results for the LASSO. In our recent work, we derived such results for the LASSO and broader classes of penalties, including non-separable ones such as SLOPE, within the standard linear model. We now extend this analysis to general loss functions, including, for example, robust regression with Huber and quantile loss functions, as well as generalized linear models from the exponential family. The main contribution of the paper is the development of an asymptotic framework for pattern convergence of regularized M-estimators under general loss functions that satisfy a suitable stochastic differentiability condition. The proofs rely on tools from empirical process theory, including Donsker classes and VC-dimension techniques.