With growing interest to use black‐box machine learning for complex data with many feature variables, it is critical to obtain a prediction model that only depends on a small set of features to maximize generalizability. Therefore, feature selection remains to be an important and challenging problem in modern applications. Most of the existing methods for feature selection are based on either parametric or semiparametric models, so the resulting performance can severely suffer from model misspecification when high‐order nonlinear interactions among the features are present. A very limited number of approaches for nonparametric feature selection were proposed, but they are computationally intensive and may not even converge. In this paper, we propose a novel and computationally efficient approach for nonparametric feature selection in the regression field based on a tensor product kernel function over the feature space. The importance of each feature is governed by a parameter in the kernel function that can be efficiently computed iteratively from a modified alternating direction method of multipliers algorithm. We prove the oracle selection property of the proposed method. Finally, we demonstrate the superior performance of our approach compared with the existing methods via simulation studies and application to the prediction of Alzheimer's disease.

中文翻译：

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具有正则化张量积核的稀疏非参数回归


随着对具有许多特征变量的复杂数据使用黑盒机器学习的兴趣日益浓厚，获得仅依赖于一小组特征以最大化泛化性的预测模型至关重要。因此，特征选择仍然是现代应用中一个重要且具有挑战性的问题。大多数现有的特征选择方法都基于参数或半参数模型，因此当特征之间存在高阶非线性相互作用时，最终的性能可能会严重受到模型错误指定的影响。提出了非常有限的非参数特征选择方法，但它们的计算量很大，甚至可能无法收敛。在本文中，我们提出了一种新颖且计算高效的方法，用于基于特征空间上的张量积核函数的回归领域中的非参数特征选择。每个特征的重要性由核函数中的参数控制，该参数可以通过改进的乘法器算法的交替方向方法迭代地有效地计算。我们证明了所提出方法的预言机选择属性。最后，我们通过模拟研究和应用于阿尔茨海默氏病的预测，证明了我们的方法与现有方法相比的优越性能。