A Direct Formulation for Sparse Pca Using Semidefinite Programming

12 Pages Posted: 13 Jul 2004

See all articles by Alexandre d'Aspremont

Alexandre d'Aspremont

Princeton University - Department of Operations Research and Financial Engineering

Laurent El Ghaoui

University of California, Berkeley - Department of Electrical Engineering & Computer Sciences (EECS)

Michael I. Jordan

University of California, Berkeley - Department of Electrical Engineering & Computer Sciences (EECS)

Gert R. Lanckriet

University of California, Berkeley - Computer Science Division

Date Written: June 2004

Abstract

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

d'Aspremont, Alexandre and El Ghaoui, Laurent and Jordan, Michael I. and Lanckriet, Gert R., A Direct Formulation for Sparse Pca Using Semidefinite Programming (June 2004). Available at SSRN: https://ssrn.com/abstract=563524 or http://dx.doi.org/10.2139/ssrn.563524

Alexandre D'Aspremont (Contact Author)

Princeton University - Department of Operations Research and Financial Engineering ( email )

Princeton, NJ 08544
United States

Laurent El Ghaoui

University of California, Berkeley - Department of Electrical Engineering & Computer Sciences (EECS) ( email )

Berkeley, CA 94720-1712
United States

Michael I. Jordan

University of California, Berkeley - Department of Electrical Engineering & Computer Sciences (EECS) ( email )

Berkeley, CA 94720-1712
United States

Gert R. Lanckriet

University of California, Berkeley - Computer Science Division ( email )

Berkeley, CA 94720-1712
United States

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