Penalized linear regression is of fundamental importance in high-dimensional statistics and has been routinely used to regress a response on a high-dimensional set of predictors. In many scientific applications, there exists external information that encodes the predictive power and sparsity structure of the predictors. In this article, we propose the Structure Adaptive Elastic-Net (SA-Enet), which provides a new framework for incorporating potentially useful side information into a penalized regression. The basic idea is to translate the external information into different penalization strengths for the regression coefficients. We particularly focus on group and covariate-dependent structures and study the risk properties of the resulting estimator. To this, we generalize the state evolution framework recently introduced for the analysis of the approximate message-passing algorithm to the SA-Enet framework. We show that the finite sample risk of the SA-Enet estimator is consistent with the theoretical risk predicted by the state evolution equation. Our theory suggests that the SA-Enet with an informative group or covariate structure can outperform the Lasso, Adaptive Lasso, Sparse Group Lasso, Feature-weighted Elastic-Net, and Graper. This evidence is further confirmed in our numerical studies. We also demonstrate the usefulness and the superiority of our method for leukemia data from molecular biology and precision medicine.
翻译:刑事线性回归在高维统计中具有根本重要性,并被例行用于倒退对高维预测器的响应。在许多科学应用中,存在外部信息,将预测器的预测力和广度结构编码。在本篇文章中,我们提议结构适应性 Elastic-Net (SA-Enet), 提供一个新的框架, 将潜在有用的侧面信息纳入惩罚性回归。 基本想法是将外部信息转化为对回归系数的不同惩罚力。 我们特别侧重于组和共变依赖结构,并研究由此产生的估计器的风险特性。 为此,我们将最近为分析近似信息传播算法而引入的国家进化框架推广到 SA- Enet 框架。 我们表明, SA- Enet 估计器的有限抽样风险与国家进化等式预测的理论风险一致。 我们的理论认为, 具有信息组或共变式结构的SA- Enet 能够超越激光索、调控性激光索、Sprassing Grouply 结构, 并研究由此产生的激光、精度组、精度精度精度精度精度精度模型研究中, 也展示了我们精度的精度的精度和精度数据方法。 我们的精度的精度和精度的精度数据。