DSPCA

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.


References in zbMATH (referenced in 34 articles )

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  1. Hatzopoulos, P.; Haberman, S.: Modeling trends in cohort survival probabilities (2015)
  2. Wang, Yang; Wu, Qiang: Sparse PCA by iterative elimination algorithm (2012)
  3. Anaya-Izquierdo, Karim; Critchley, Frank; Vines, Karen: Orthogonal simple component analysis: a new, exploratory approach (2011)
  4. d’Aspremont, Alexandre: Identifying small mean-reverting portfolios (2011)
  5. d’Aspremont, Alexandre; El Ghaoui, Laurent: Testing the nullspace property using semidefinite programming (2011)
  6. Hardoon, David R.; Shawe-Taylor, John: Sparse canonical correlation analysis (2011)
  7. Hatzopoulos, P.; Haberman, S.: A dynamic parameterization modeling for the age-period-cohort mortality (2011)
  8. Iyengar, G.; Phillips, D.J.; Stein, C.: Approximating semidefinite packing programs (2011)
  9. Luss, Ronny; Teboulle, Marc: Convex approximations to sparse PCA via Lagrangian duality (2011)
  10. Rinaldi, F.: Concave programming for finding sparse solutions to problems with convex constraints (2011)
  11. Sagnol, Guillaume: A class of semidefinite programs with rank-one solutions (2011)
  12. Sriperumbudur, Bharath K.; Torres, David A.; Lanckriet, Gert R.G.: A majorization-minimization approach to the sparse generalized eigenvalue problem (2011)
  13. Zhou, Tianyi; Tao, Dacheng; Wu, Xindong: Manifold elastic net: a unified framework for sparse dimension reduction (2011)
  14. Duong, Thanh D.X.; Duong, Vu N.: Principal component analysis with weighted sparsity constraint (2010)
  15. Gillis, Nicolas; Glineur, François: Using underapproximations for sparse nonnegative matrix factorization (2010)
  16. Iyengar, Garud; Phillips, David J.; Stein, Cliff: Feasible and accurate algorithms for covering semidefinite programs (2010)
  17. Journée, M.; Bach, F.; Absil, P.-A.; Sepulchre, R.: Low-rank optimization on the cone of positive semidefinite matrices (2010)
  18. Journée, Michel; Nesterov, Yurii; Richtárik, Peter; Sepulchre, Rodolphe: Generalized power method for sparse principal component analysis (2010)
  19. Mairal, Julien; Bach, Francis; Ponce, Jean; Sapiro, Guillermo: Online learning for matrix factorization and sparse coding (2010)
  20. Ruiz-Torrubiano, Rubén; García-Moratilla, Sergio; Suárez, Alberto: Optimization problems with cardinality constraints (2010)

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