论文标题
基于弹性网的稀疏主成分分析的理论保证
Theoretical Guarantees for Sparse Principal Component Analysis based on the Elastic Net
论文作者
论文摘要
稀疏主成分分析（SPCA）在高维数据分析中广泛用于降低维度和特征提取。尽管过去二十年来有许多方法论和理论发展，但Zou，Hastie＆Tibshirani（2006）提出的流行的SPCA算法的理论保证仍然未知。本文旨在解决这一关键差距。我们首先重新访问Zou等人的SPCA算法。 （2006年），介绍我们的实施。我们还研究了Zou等人的SPCA算法的计算更有效的变体。 （2006年）可以被视为SPCA的限制案例。我们提供了算法的固定点的收敛保证，并证明，在稀疏的尖峰协方差模型下，这两种算法都可以在轻度的规律性条件下始终如一地恢复主要子空间。我们表明，他们的估计误差界限与现有作品的最佳可用界限或最小值速率与某些对数因素相匹配。此外，我们证明了两种算法在数值研究中的竞争性数值性能。
Sparse principal component analysis (SPCA) is widely used for dimensionality reduction and feature extraction in high-dimensional data analysis. Despite many methodological and theoretical developments in the past two decades, the theoretical guarantees of the popular SPCA algorithm proposed by Zou, Hastie & Tibshirani (2006) are still unknown. This paper aims to address this critical gap. We first revisit the SPCA algorithm of Zou et al. (2006) and present our implementation. We also study a computationally more efficient variant of the SPCA algorithm in Zou et al. (2006) that can be considered as the limiting case of SPCA. We provide the guarantees of convergence to a stationary point for both algorithms and prove that, under a sparse spiked covariance model, both algorithms can recover the principal subspace consistently under mild regularity conditions. We show that their estimation error bounds match the best available bounds of existing works or the minimax rates up to some logarithmic factors. Moreover, we demonstrate the competitive numerical performance of both algorithms in numerical studies.
