A direct formulation for sparse PCA using semidefinite programming
<|reference_start|>A direct formulation for sparse PCA using semidefinite programming: 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.<|reference_end|>
@article{d'aspremont2004a,
    title={A direct formulation for sparse PCA using semidefinite programming},
    author={Alexandre d'Aspremont, Laurent El Ghaoui, Michael I. Jordan, Gert R.
  G. Lanckriet},
    journal={arXiv preprint arXiv:cs/0406021},
    year={2004},
    number={Working paper: UCB//CSD-04-1330},
    archivePrefix={arXiv},
    eprint={cs/0406021},
    primaryClass={cs.CE}
}