Part of Advances in Neural Information Processing Systems 17 (NIPS 2004)
Alexandre D'aspremont, Laurent E. Ghaoui, Michael I. Jordan, Gert R. Lanckriet
We examine the problem of approximating, in the Frobenius-norm sense, a positive, semidefinite symmetric matrix by a rank-one matrix, with an upper bound on the cardinality of its eigenvector. The problem arises in the decomposition of a covariance matrix into sparse factors, and has wide applications ranging from biology to finance. We use a modifica- tion of the classical variational representation of the largest eigenvalue of a symmetric matrix, where cardinality is constrained, and derive a semidefinite programming based relaxation for our problem.