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.

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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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