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

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

2006-01-18

arxiv.org/abs/math/0601448v2

Mathematics

Optimization and Control

13 pages, 3 figures This new version corresponds to an extensive revision of the earlier version

Scientific paper

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).

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