Generalized additive model is a powerful statistical learning and predictive modeling tool that has been applied in a wide range of applications. The need of high-dimensional additive modeling is eminent in the context of dealing with high throughput data such as genetics data analysis. In this article, we studied a two-step selection and estimation method for ultrahigh-dimensional generalized additive models. The first step applies group lasso on the expanded bases of the functions. With high probability this selects all nonzero functions without having too much over selection. The second step uses adaptive group lasso with any initial estimators, including the group lasso estimator, that satisfies some regular conditions. The adaptive group lasso estimator is shown to be selection consistent with improved convergence rates. Tuning parameter selection is also discussed and shown to select the true model consistently under generalized information criterion procedure. The theoretical properties are supported by extensive numerical study.

中文翻译：

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超高维广义加法模型：统一理论与方法

广义加法模型是一种强大的统计学习和预测建模工具，已在广泛的应用中得到应用。在处理高通量数据（如遗传数据分析）的背景下，高维加性建模的需求非常突出。在本文中，我们研究了一种超高维广义加性模型的两步选择和估计方法。第一步在函数的扩展基础上应用组套索。这很有可能会选择所有非零函数，而不会过度选择。第二步使用自适应组套索和任何初始估计器，包括满足一些常规条件的组套索估计器。自适应组套索估计器被证明是与改进的收敛速度一致的选择。还讨论并显示了调整参数选择以在广义信息准则程序下一致地选择真实模型。理论性质得到广泛的数值研究的支持。