Principal component analysis (PCA) has been widely used as an effective technique for feature extraction and dimension reduction. In the High Dimension Low Sample Size setting, one may prefer modified principal components, with penalized loadings, and automated penalty selection by implementing model selection among these different models with varying penalties. The earlier work (Zou et al. in J Comput Graph Stat 15(2):265–286, 2006; Gaynanova et al. in J Comput Graph Stat 26(2):379–387, 2017) has proposed penalized PCA, indicating the feasibility of model selection in \(\ell _2\)-penalized PCA through the solution path of Ridge regression, however, it is extremely time-consuming because of the intensive calculation of matrix inverse. In this paper, we propose a fast model selection method for penalized PCA, named approximated gradient flow (AgFlow), which lowers the computation complexity through incorporating the implicit regularization effect introduced by (stochastic) gradient flow (Ali et al. in: The 22nd international conference on artificial intelligence and statistics, pp 1370–1378, 2019; Ali et al. in: International conference on machine learning, 2020) and obtains the complete solution path of \(\ell _2\)-penalized PCA under varying \(\ell _2\)-regularization. We perform extensive experiments on real-world datasets. AgFlow outperforms existing methods (Oja and Karhunen in J Math Anal Appl 106(1):69–84, 1985; Hardt and Price in: Advances in neural information processing systems, pp 2861–2869, 2014; Shamir in: International conference on machine learning, pp 144–152, PMLR, 2015; and the vanilla Ridge estimators) in terms of computation costs.

主成分分析（PCA）已被广泛用作特征提取和降维的有效技术。在 High Dimension Low Sample Size 设置中，人们可能更喜欢修改后的主成分、惩罚加载和通过在这些具有不同惩罚的不同模型中实现模型选择的自动惩罚选择。早期的工作（Zou 等人在 J Comput Graph Stat 15(2):265-286, 2006 中；Gaynanova 等人在 J Comput Graph Stat 26(2):379-387, 2017 中）提出了惩罚 PCA，表明\(\ell _2\)中模型选择的可行性- 通过岭回归的求解路径惩罚PCA，但是由于矩阵求逆的计算量很大，所以非常耗时。在本文中，我们提出了一种惩罚 PCA 的快速模型选择方法，称为近似梯度流 ( AgFlow )，它通过结合（随机）梯度流引入的隐式正则化效应来降低计算复杂度（Ali et al. in: The 22nd人工智能与统计国际会议，pp 1370–1378，2019；Ali et al. in: International Conference on machine learning, 2020) 并获得了\(\ell _2\) -penalized PCA 在不同\( \ell _2\) - 正则化。我们对真实世界的数据集进行了广泛的实验。AgFlow优于现有方法（Oja 和 Karhunen 在 J Math Anal Appl 106(1):69-84，1985 年；Hardt 和 Price 在：神经信息处理系统的进展，pp 2861-2869，2014 年；Shamir 在：国际机器会议学习，第 144-152 页，PMLR，2015 年；以及 vanilla Ridge 估计器）在计算成本方面。