Sparsistency and agnostic inference in sparse PCA
Sparsistency and agnostic inference in sparse PCA
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DOI：
10.1214/14-aos1273
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发表时间：
2014-01
影响因子：
4.5
通讯作者：
Jing Lei;Vincent Q. Vu
中科院分区：
文献类型：
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作者：
Jing Lei;Vincent Q. Vu
稀疏“真相”的存在一直是稀疏 PCA 理论分析中的一个恒定假设，并且通常隐含在其方法发展中。这自然会引发关于稀疏 PCA 方法的属性以及它们如何依赖于稀疏性假设的问题。如果假设真相是稀疏的，在什么条件下可以一致地选择相关变量？在不假设稀疏且唯一的事实的情况下，如何看待稀疏 PCA 的结果？我们通过研究最近提出的 Fantope 投影和选择（FPS）方法在高维环境中的属性来回答这些问题。我们的结果为 FPS 估计器的稀疏性提供了一般的充分条件。这些条件很弱，并且可以在已知其他估计器失败的情况下成立。另一方面，在不假设稀疏性或可识别性的情况下，我们表明 FPS 提供了一种稀疏的线性降维变换，在最大化预测协方差方面接近于最佳可能。
The presence of a sparse "truth" has been a constant assumption in the theoretical analysis of sparse PCA and is often implicit in its methodological development. This naturally raises questions about the properties of sparse PCA methods and how they depend on the assumption of sparsity. Under what conditions can the relevant variables be selected consistently if the truth is assumed to be sparse? What can be said about the results of sparse PCA without assuming a sparse and unique truth? We answer these questions by investigating the properties of the recently proposed Fantope projection and selection (FPS) method in the high-dimensional setting. Our results provide general sufficient conditions for sparsistency of the FPS estimator. These conditions are weak and can hold in situations where other estimators are known to fail. On the other hand, without assuming sparsity or identifiability, we show that FPS provides a sparse, linear dimension-reducing transformation that is close to the best possible in terms of maximizing the predictive covariance.