Generalized power method for sparse principal component analysis
AbstractIn this paper we develop a new approach to sparse principal component analysis (sparse PCA). We propose two single-unit and two block optimization formulations of the sparse PCA problem, aimed at extracting a single sparse dominant principal component of a data matrix, or more components at once, respectively. While the initial formulations involve nonconvex functions, and are therefore computationally intractable, we rewrite them into the form of an optimization program involving maximization of a convex function on a compact set. The dimension of the search space is decreased enormously if the data matrix has many more columns (variables) than rows. We then propose and analyze a simple gradient method suited for the task. It appears that our algorithm has best convergence properties in the case when either the objective function or the feasible set are strongly convex, which is the case with our single-unit formulations and can be enforced in the block case. Finally, we demonstrate numerically on a set of random and gene expression test problems that our approach outperforms existing algorithms both in quality of the obtained solution and in computational speed.
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Bibliographic InfoPaper provided by Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) in its series CORE Discussion Papers with number 2008070.
Date of creation: 01 Nov 2008
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sparse PCA; power method; gradient ascent; strongly convex sets; block algorithms.;
This paper has been announced in the following NEP Reports:
- NEP-ALL-2009-04-05 (All new papers)
- NEP-CMP-2009-04-05 (Computational Economics)
- NEP-ECM-2009-04-05 (Econometrics)
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