On the Quality of a Semidefinite Programming Bound for Sparse Principal Component Analysis

Laurent El Ghaoui (elghaouieecs.berkeley.edu)

Abstract: We examine the problem of approximating a positive, semidefinite matrix $\Sigma$ by a dyad $xx^T$, with a penalty on the cardinality of the vector $x$. This problem arises in sparse principal component analysis, where a decomposition of $\Sigma$ involving sparse factors is sought. We express this hard, combinatorial problem as a maximum eigenvalue problem, in which we seek to maximize, over a box, the largest eigenvalue of a symmetric matrix that is linear in the variables. This representation allows to use the techniques of robust optimization, to derive a bound based on semidefinite programming. The quality of the bound is investigated using a technique inspired by Nemirovski and Ben-Tal (2002).

Keywords: Sparse Principal Component Analysis, Semidefinite Programming, Quality of Relaxation, Unsupervised Machine Learning

Category 1: Robust Optimization

Category 2: Applications -- Science and Engineering (Statistics )

Category 3: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Citation: unpublished

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Entry Submitted: 03/10/2006
Entry Accepted: 03/10/2006
Entry Last Modified: 03/10/2006

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