%0 Journal Article
%T On the Quality of a Semidefinite Programming Bound for Sparse Principal Component Analysis
%A Laurent El Ghaoui
%J Mathematics
%D 2006
%I arXiv
%X We examine the problem of approximating a positive, semidefinite matrix $\Sigma$ by a dyad $xx^T$, with a penalty on the cardinality of the vector $x$. This problem arises in sparse principal component analysis, where a decomposition of $\Sigma$ involving sparse factors is sought. We express this hard, combinatorial problem as a maximum eigenvalue problem, in which we seek to maximize, over a box, the largest eigenvalue of a symmetric matrix that is linear in the variables. This representation allows to use the techniques of robust optimization, to derive a bound based on semidefinite programming. The quality of the bound is investigated using a technique inspired by Nemirovski and Ben-Tal (2002).
%U http://arxiv.org/abs/math/0601448v2