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Optimal Feature Selection in High-Dimensional Discriminant Analysis

机译：高维判别分析中的最佳特征选择

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摘要

We consider the high-dimensional discriminant analysis problem. For this problem, different methods have been proposed and justified by establishing exact convergence rates for the classification risk, as well as the convergence results to the discriminative rule. However, sharp theoretical analysis for the variable selection performance of these procedures have not been established, even though model interpretation is of fundamental importance in scientific data analysis. This paper bridges the gap by providing sharp sufficient conditions for consistent variable selection using the sparse discriminant analysis. Through careful analysis, we establish rates of convergence that are significantly faster than the best known results and admit an optimal scaling of the sample size , dimensionality , and sparsity level in the high-dimensional setting. Sufficient conditions are complemented by the necessary information theoretic limits on the variable selection problem in the context of high-dimensional discriminant analysis. Exploiting a numerical equivalence result, our method also establish the optimal results for the ROAD estimator and the sparse optimal scoring estimator. Furthermore, we analyze an exhaustive search procedure, whose performance serves as a benchmark, and show that it is variable selection consistent under weaker conditions. Extensive simulations demonstrating the sharpness of the bounds are also provided.
机译：我们考虑高维判别分析问题。对于这个问题，已经提出了不同的方法并通过建立分类风险的精确收敛率以及对判别规则的收敛结果来证明其合理性。但是，尽管模型解释在科学数据分析中具有根本重要性，但尚未对这些程序的变量选择性能进行清晰的理论分析。本文通过使用稀疏判别分析为一致的变量选择提供了尖锐的充分条件，从而弥合了差距。通过仔细的分析，我们确定收敛速度明显快于最著名的结果，并在高维环境中接受了样本大小，维数和稀疏度级别的最佳缩放。在高维判别分析中，足够的条件由变量选择问题的必要信息理论限制所补充。利用数值等价结果，我们的方法还为ROAD估计器和稀疏最优评分估计器建立了最佳结果。此外，我们分析了穷举搜索过程，该搜索过程的性能作为基准，并表明它在较弱条件下的变量选择是一致的。还提供了证明边界清晰度的广泛模拟。

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