Approximation bounds for sparse principal component analysis

• Published in: Mathematical programming Vol. 148; no. 1-2; pp. 89 - 110
• Main Authors: d’Aspremont, Alexandre, Bach, Francis, Ghaoui, Laurent El
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2014; Springer Nature B.V; Springer Verlag
• Subjects: Algorithms; Analysis; Approximation; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Computer science; Constants; Data points; Eigenvalues; Full Length Paper; Gaussian; Hypotheses; Hypothesis testing; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Optimization and Control; Principal component analysis; Principal components analysis; Ratios; Semidefinite programming; Sparsity; Statistics; Studies; Theoretical; 90C22; 90C27; 62H25
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-014-0751-7

We produce approximation bounds on a semidefinite programming relaxation for sparse principal component analysis. The sparse maximum eigenvalue problem cannot be efficiently approximated up to a constant approximation ratio, so our bounds depend on the optimum value of the semidefinite relaxation: the higher this value, the better the approximation. In particular, these bounds allow us to control approximation ratios for tractable statistics in hypothesis testing problems where data points are sampled from Gaussian models with a single sparse leading component.