  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 modification 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. We also discuss Nesterov's smooth
minimization technique applied to the SDP arising in the direct sparse PCA
method.
