Eigenvalue Maximization in Sparse PCA

We examine the problem of approximating a positive, semidefinite matrix Σ by a dyad xxT , with a penalty on the cardinality of the vector x. This problem arises in the sparse principal component analysis problem, where a decomposition of Σ 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 primalization technique inspired by Nemirovski and Ben-Tal (2002). Notation. The notation 1 denotes the vector of ones (with size inferred from context), while Card(x) denotes the cardinality of a vector x (number of non-zero elements). We denote by ei the unit vectors of R . For a n × n matrix X, X 0 means X is symmetric and positive semi-definite. The notation B+, for a symmetric matrix B, denotes the matrix obtained from B by replacing negative eigenvalues by 0. Throughout, the symbol E refers to expectations taken with respect to the normal Gaussian distribution.