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Generalized power method for sparse principal component analysis

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  • Journée, Michel

    (---)

  • Nesterov, Yurii

    (Université catholique de Louvain (UCL). Center for Operations Research and Econometrics (CORE))

  • Richtarik, Peter

    (Université catholique de Louvain (UCL). Center for Operations Research and Econometrics (CORE))

  • Sepulchre, Rodolphe

Abstract

In 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.

Suggested Citation

  • Journée, Michel & Nesterov, Yurii & Richtarik, Peter & Sepulchre, Rodolphe, 2008. "Generalized power method for sparse principal component analysis," CORE Discussion Papers 2008070, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  • Handle: RePEc:cor:louvco:2008070
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    File URL: http://www.uclouvain.be/cps/ucl/doc/core/documents/coredp2008_70.pdf
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    Keywords

    sparse PCA; power method; gradient ascent; strongly convex sets; block algorithms.;

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