# A Direct Formulation for Sparse PCA Using Semidefinite Programming

### Alexandre d'Aspremont, Laurent El Ghaoui, Michael I. Jordan and Gert R. G. Lanckriet

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-04-1330

June 2004

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2004/CSD-04-1330.pdf

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. The method has complexity
*O*(
*n*^4 square root of log(
*n*) / epsilon), where
*n* is the size of the underlying covariance matrix, and epsilon is the desired absolute accuracy on the optimal value of the problem.

BibTeX citation:

@techreport{d'Aspremont:CSD-04-1330, Author = {d'Aspremont, Alexandre and El Ghaoui, Laurent and Jordan, Michael I. and Lanckriet, Gert R. G.}, Title = {A Direct Formulation for Sparse PCA Using Semidefinite Programming}, Institution = {EECS Department, University of California, Berkeley}, Year = {2004}, Month = {Jun}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2004/5534.html}, Number = {UCB/CSD-04-1330}, 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. The method has complexity <i>O</i>(<i>n</i>^4 square root of log(<i>n</i>) / epsilon), where <i>n</i> is the size of the underlying covariance matrix, and epsilon is the desired absolute accuracy on the optimal value of the problem.} }

EndNote citation:

%0 Report %A d'Aspremont, Alexandre %A El Ghaoui, Laurent %A Jordan, Michael I. %A Lanckriet, Gert R. G. %T A Direct Formulation for Sparse PCA Using Semidefinite Programming %I EECS Department, University of California, Berkeley %D 2004 %@ UCB/CSD-04-1330 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2004/5534.html %F d'Aspremont:CSD-04-1330