Sparse common component analysis for multiple high-dimensional datasets via noncentered principal component analysis

There is currently much discussion about the analysis of multiple datasets from different groups, among which especially identifying a common basic structure of multiple groups has drawn a large amount of attention. In order to identify a common basic structure, common component analysis (CCA) was proposed by generalizing techniques for principal component analysis (PCA); i.e., CCA becomes standard PCA when applied to only one dataset. Although CCA can identify the common structure of multiple datasets, which cannot be extracted by standard PCA, CCA suffers from the following drawbacks. The common components are estimated as linear combinations of all variables, and thus it is difficult to interpret the identified common components. The fully dense loadings lead to erroneous results in CCA, because noisy features are inevitably included in datasets. To address these issues, we incorporate sparsity into CCA, and propose a novel strategy for sparse common component analysis based on L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{1}$$\end{document}-type regularized regression modeling. We focus CCA which is formulated as the eigenvalue decomposition (EVD) of a Gram matrix (i.e., common loadings of multiple datasets can be estimated by EVD of a Gram matrix), and it can be performed by Singular value decomposition of a square root of the Gram matrix. We then propose sparse common component analysis based on sparse PCA to estimate sparse common loadings of multiple datasets. We also propose an algorithm to estimate sparse common loadings of multiple datasets. The proposed method can not only identify a common subspace but also select crucial common-features for multiple groups. Monte Carlo simulations and real-data analysis are conducted to examine the efficiency of the proposed sparse CCA. We observe from the numerical studies that our strategies can incorporate sparsity into the common loading estimation and efficiently recover a sparse common structure efficiently in multiple dataset analysis.