A direct formulation for sparse PCA using semidefinite programming

Details A direct formulation for sparse PCA using semidefinite programming A direct formulation for sparse PCA using semidefinite programming

2004-06-16

arxiv.org/abs/cs/0406021v3

Computer Science

Computational Engineering, Finance, and Science

Final version, to appear in SIAM review

Scientific paper

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.

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