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

Laurent El Ghaoui

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arXiv · math/0601448

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

Laurent El Ghaoui

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Abstract

We examine the problem of approximating a positive, semidefinite matrix $Σ$ 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 $Σ$ 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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Laurent El Ghaoui. 2006-02-03. On the Quality of a Semidefinite Programming Bound for Sparse Principal Component Analysis. https://arxiv.org/abs/math/0601448

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