Journal of Global Optimization

, Volume 70, Issue 4, pp 843–858 | Cite as

Tensor maximal correlation problems

  • Anwa Zhou
  • Xin Zhao
  • Jinyan Fan
  • Yanqin Bai


This paper studies the tensor maximal correlation problem, which aims at optimizing correlations between sets of variables in many statistical applications. We reformulate the problem as an equivalent polynomial optimization problem, by adding the first order optimality condition to the constraints, then construct a hierarchy of semidefinite relaxations for solving it. The global maximizers of the problem can be detected by solving a finite number of such semidefinite relaxations. Numerical experiments show the efficiency of the proposed method.


Tensor maximal correlation problems Polynomial optimization Lasserre relaxation Semidefinite program 

Mathematics Subject Classification

62H20 65K05 90C22 


  1. 1.
    Chu, M., Watterson, J.: On a multivariate eigenvalue problem, Part I: algebraic theory and a power method. SIAM J. Sci. Comput. 14, 1089–1106 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cui, C., Dai, Y., Nie, J.: All real eigenvalues of symmetric tensors. SIAM J. Matrix Anal. Appl. 35, 1582–1601 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Curto, R., Fialkow, L.: Truncated K-moment problems in several variables. J. Operator Theory 54, 189–226 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Demmel, J.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)CrossRefzbMATHGoogle Scholar
  5. 5.
    Golub, G .H., Van Loan, C .F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)zbMATHGoogle Scholar
  6. 6.
    Hanafi, M., Ten Berge, J.M.F.: Global optimality of the successive Maxbet algorithm. Psychometrika 68, 97–103 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Helton, J.W., Nie, J.: A semidefinite approach for truncated K-moment problems. Found. Comput. Math. 12, 851–881 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Henrion, D., Lasserre, J.: Detecting global optimality and extracting solutions in GloptiPoly, Positive polynomials in control, Lecture Notes in Control and Inform. Sci. Springer, Berlin, 312 , pp. 293–310 (2005)Google Scholar
  9. 9.
    Henrion, D., Lasserre, J., Loefberg, J.: GloptiPoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24, 761–779 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Horst, P.: Relations among m sets of measures. Psychometrika 26, 129–149 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hotelling, H.: The most predictable criterion. J. Educ. Pyschol. 26, 139–142 (1935)CrossRefGoogle Scholar
  12. 12.
    Hotelling, H.: Relations between two sets of variates. Biometrika 28, 321–377 (1936)CrossRefzbMATHGoogle Scholar
  13. 13.
    Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lasserre, J.B.: Moments, Positive Polynomials and Their Applications. Imperial College Press, London (2009)CrossRefGoogle Scholar
  15. 15.
    Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry, Vol. 149 of IMA Volumes in Mathematics and its Applications, pp. 157–270. Springer, New York (2009)Google Scholar
  16. 16.
    Nie, J., Demmel, J., Sturmfels, B.: Minimizing polynomials via sum of squares over the gradient ideal. Math. Program., Ser. A 106, 587–606 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nie, J.: Certifying convergence of Lasserre’s hierarchy via flat truncation. Math. Program., Ser. A 142, 485–510 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nie, J.: Polynomial optimization with real varieties. SIAM J. Optim. 23, 1634–1646 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nie, J.: An exact Jacobian SDP relaxation for polynomial optimization. Math. Program. 137, 225–255 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nie, J.: Optimality conditions and finite convergence of Lasserre’s hierarchy. Math. Program., Ser. A 146, 97–121 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nie, J.: The \(A\)-truncated K-moment problem. Found. Comput. Math. 14, 1243–1276 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nie, J.: Linear optimization with cones of moments and nonnegative polynomials. Math. Program., Ser. B 153, 247–274 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Putinar, M.: Positive polynomials on compact semi-algebraic sets. Ind. Aniv. Math. J. 42, 969–984 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sigg, C., Fischer, B., Ommer, B., Roth, V., Buhmann, J.: Nonnegative CCA for audiovisual source separation. In: IEEE International Workshop on Machine Learning for Signal Processing 07, IEEE (2007)Google Scholar
  25. 25.
    Sturm, J.F.: SeDuMi 1.02: a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw., 11 & 12, 625–653 (1999).
  26. 26.
    Timm, N.H.: Applied Multivariate Analysis. Springer, New York (2002)zbMATHGoogle Scholar
  27. 27.
    Van de Geer, J.R.: Linear relations among k sets of variables. Psychometrika 49, 79–94 (1984)CrossRefGoogle Scholar
  28. 28.
    Zhang, L., Liao, L., Sun, L.: Towards the global solution of the maximal correlation problem. J. Glob. Optim. 49, 91–107 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zhang, L., Chu, M.: Computing absolute maximum correlation. IMA J. Numer. Anal. 32, 163–184 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Zhang, L., Liao, L.: An alternating variable method for the maximal correlation problem. J. Glob. Optim. 54, 199–218 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiPeople’s Republic of China
  2. 2.School of Mathematical SciencesShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
  3. 3.School of Mathematical Sciences, and MOE-LSCShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China

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