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Optimality conditions for penalized sparse PCA
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DOI：10.1007/s11081-025-10021-0.png)
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Abstract
En 中文
This paper establishes the theoretical foundations of an alternating optimization scheme for penalized sparse principal component analysis (PCA) focusing on variance maximization. We provide a theoretical foundation for the optimality of solutions derived from this widely used algorithm, addressing a gap in the current literature where empirical results often lack theoretical support. We show the algorithm's success when the dataset's covariance matrix is positive definite. Additionally, we characterize sparsity-inducing penalties and examine the use of various ones, including the & ell;1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \ell _1$$\end{document} -norm, SCAD, and & ell;0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _0$$\end{document} -norm. We conduct numerical experiments to evaluate standard metrics, such as explained variance, number of iterations, and computational time.
Keywords:
Sparse PCA
Penalties
Optimality conditions
Thresholding operators
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