An exact approach to sparse principal component analysis
In sparse principal components analysis (sPCA) one looks for principal directions \(\beta\) in the set of vectors with a limited number of nonzero entries. E.g., the first sPCA component of the covariance matrix \(Z\) is defined as the solution of the optimization problem \(\max_\beta \beta' Z\beta\) under the constraints \(\beta'\beta=1\) and \(| \beta| _0\leq k_1\), where \(| \beta| _0\) is the number of nonzero entries in \(\beta\), and \(k_1\) is a fixed number (the degree of sparsity). The author proposes a new branch-and-bound algorithm for the exact solution of this problem and describes its application to the calculation of many (orthogonal or uncorrelated) sPCA components. The choice of the degree of sparsity is discussed. Results of simulations and real data analyses are presented.
scientific article; zbMATH DE number 5946106