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In this paper, we consider the sparse eigenvalue problem wherein the goal isto obtain a sparse solution to the generalized eigenvalue problem. We achievethis by constraining the cardinality of the solution to the generalizedeigenvalue problem and obtain sparse principal component analysis (PCA), sparsecanonical correlation analysis (CCA) and sparse Fisher discriminant analysis(FDA) as special cases. Unlike the $\ell_1$-norm approximation to thecardinality constraint, which previous methods have used in the context ofsparse PCA, we propose a tighter approximation that is related to the negativelog-likelihood of a Student's t-distribution. The problem is then framed as ad.c. (difference of convex functions) program and is solved as a sequence ofconvex programs by invoking the majorization-minimization method. The resultingalgorithm is proved to exhibit \emph{global convergence} behavior, i.e., forany random initialization, the sequence (subsequence) of iterates generated bythe algorithm converges to a stationary point of the d.c. program. Theperformance of the algorithm is empirically demonstrated on both sparse PCA(finding few relevant genes that explain as much variance as possible in ahigh-dimensional gene dataset) and sparse CCA (cross-language documentretrieval and vocabulary selection for music retrieval) applications.

A D.C. Programming Approach to the Sparse Generalized Eigenvalue Problem