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An alternating variable method for the maximal correlation problem


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  • Lei-Hong Zhang


  • Li-Zhi Liao


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    The maximal correlation problem (MCP) aiming at optimizing correlations between sets of variables plays an important role in many areas of statistical applications. Up to date, algorithms for the general MCP stop at solutions of the multivariate eigenvalue problem (MEP), which serves only as a necessary condition for the global maxima of the MCP. For statistical applications, the global maximizer is quite desirable. In searching the global solution of the MCP, in this paper, we propose an alternating variable method (AVM), which contains a core engine in seeking a global maximizer. We prove that (i) the algorithm converges globally and monotonically to a solution of the MEP, (ii) any convergent point satisfies a global optimal condition of the MCP, and (iii) whenever the involved matrix A is nonnegative irreducible, it converges globally to the global maximizer. These properties imply that the AVM is an effective approach to obtain a global maximizer of the MCP. Numerical testings are carried out and suggest a superior performance to the others, especially in finding a global solution of the MCP. Copyright Springer Science+Business Media, LLC. 2012

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    Bibliographic Info

    Article provided by Springer in its journal Journal of Global Optimization.

    Volume (Year): 54 (2012)
    Issue (Month): 1 (September)
    Pages: 199-218

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    Handle: RePEc:spr:jglopt:v:54:y:2012:i:1:p:199-218

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    Keywords: Multivariate statistics; Canonical correlation; Maximal correlation problem; Multivariate eigenvalue problem; Power method; Gauss–Seidal method; Global maximizer; 62H20; 15A12; 65F10; 65K05;

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    1. John Geer, 1984. "Linear relations amongk sets of variables," Psychometrika, Springer, vol. 49(1), pages 79-94, March.
    2. Mohamed Hanafi & Jos Berge, 2003. "Global optimality of the successive Maxbet algorithm," Psychometrika, Springer, vol. 68(1), pages 97-103, March.
    3. Paul Horst, 1961. "Relations amongm sets of measures," Psychometrika, Springer, vol. 26(2), pages 129-149, June.
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