Abstract
Sparse principal component analysis (SPCA) has achieved great success in improving interpretable ability of the derived results and has become a powerful technique for modern data analysis. It presents that principal component can be modified to produce sparse loadings by imposing sparsity-induced penalty, which is often \(l_{1}\)-regularized constraint. In order to analyze the \(l_{1}\)-regularized sparsity-induced model, in this paper, we propose a general null space property of a matrix \(\mathbf {A}\) relative to a index set S and give a necessary and sufficient condition for the exact or approximate sparse principal components. Meanwhile, the conclusions with respect to the stable and robust situations are given in the case of exact or approximate sparse principal components, respectively.
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Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
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Acknowledgements
The authors would like to thank the anonymous reviewers and the associate editor and editor-in-chief for their constructive suggestions, which improve the manuscript significantly. This work was supported by the National Natural Science Foundation of China under the Grants No. 11771347, 91730306, 41390454.
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Han, X., Peng, J., Cui, A. et al. A General Null Space Property for Sparse Principal Component Analysis. Circuits Syst Signal Process 41, 4570–4580 (2022). https://doi.org/10.1007/s00034-022-01991-y
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DOI: https://doi.org/10.1007/s00034-022-01991-y