A Direct Formulation for Sparse Pca Using Semidefinite Programming
12 Pages Posted: 13 Jul 2004
Date Written: June 2004
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.
Keywords: PCA, semidefinite programming, relaxation, sparse
JEL Classification: C10, C44
Suggested Citation: Suggested Citation