Modern multivariate machine learning and statistical methodologies estimate parameters of interest while leveraging prior knowledge of the association between outcome variables. The methods that do allow for estimation of relationships do so typically through an error covariance matrix in multivariate regression which does not scale to other types of models. In this article we proposed the MinPEN framework to simultaneously estimate regression coefficients associated with the multivariate regression model and the relationships between outcome variables using mild assumptions. The MinPen framework utilizes a novel penalty based on the minimum function to exploit detected relationships between responses. An iterative algorithm that generalizes current state of the art methods is proposed as a solution to the non-convex optimization that is required to obtain estimates. Theoretical results such as high dimensional convergence rates, model selection consistency, and a framework for post selection inference are provided. We extend the proposed MinPen framework to other exponential family loss functions, with a specific focus on multiple binomial responses. Tuning parameter selection is also addressed. Finally, simulations and two data examples are presented to show the finite sample properties of this framework.
翻译:现代多变机器学习和统计方法在利用先前对结果变量之间关联的认识的同时,估计兴趣参数,同时利用先前对结果变量之间关联的了解。 允许估计关系的方法通常通过多变回归中的差差差共变矩阵来估计关系,该矩阵不至于扩大到其他类型的模型。 在本条中,我们提议了 MinPEN 框架,以同时估计与多变回归模型相关的回归系数和结果变量之间的关系。 MinPen 框架使用基于最低功能的新惩罚来利用所发现的反应之间的关系。 提出了一个迭代算法,将最新先进方法的当前状态概括化,作为获取估计所需的非covex优化的一种解决办法。 提供了理论结果, 如高维趋一致率、模型选择一致性和后推推法框架。 我们将拟议的 MinPen 框架扩大到其他指数家庭损失函数, 具体侧重于多个双向响应。 也述及了Tuning 参数选择。 最后, 模拟和两个数据示例展示了这个框架的有限抽样特性。