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Successive projection method for solving the unbalanced Procrustes problem

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Abstract

We present a successive projection method for solving the unbalanced Procrustes problem: given matrix A ∈ R n × n and B ∈ R n × k, n > k, minimize the residual ‖AQ − B‖F with the orthonormal constraint Q T Q = I k on the variant Q ∈ R n × k. The presented algorithm consists of solving k least squares problems with quadratic constraints and an expanded balance problem at each sweep. We give a detailed convergence analysis. Numerical experiments reported in this paper show that our new algorithm is superior to other existing methods.

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References

  1. Green B F, Goers J C. A Problem with Congruence. The Annual Meeting of the Psychometric Society, Monterey, California, 1979

  2. Ten Berge J M F, Konl D L. Orthogonal rotations to maximal agreement for two or more matrices of different column orders. Psychometrica, 1984, 49: 49–55

    Article  MATH  Google Scholar 

  3. Park H. A parallel algorithm for the unbalanced orthogonal procrustes problem. Parallel Computing, 1991, 17: 913–923

    Article  MATH  Google Scholar 

  4. Bojanczyk A W, Lutoborski A. The procrustes problem for orthogonal stiefel matrices. SIAM J Sci Comput, 1999, 21(4): 1291–1304

    Article  MathSciNet  Google Scholar 

  5. Schönemann P H. A generalized solution of the orthogonal procrustes problem. Psychometrika, 1966, 31(1): 1–10

    Article  MATH  MathSciNet  Google Scholar 

  6. Golub G H, Van Loan C F. Matrix Computations. 3nd ed. Baltimore: Johns Hopkins University Press, 1996

    Google Scholar 

  7. Chu M T, Trendafilov N T. The orthogonally constrained regression revisited. J Computat Graph Stat, 2001, 10(4): 746–771

    Article  MathSciNet  Google Scholar 

  8. Zhang Z, Huang Y. A projection method for least squares problems with a quadratic equality constraint. SIAM J Matr Anal Appl, 2003, 25(1): 188–212

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Zhang Zhenyue & Du Keqin

Authors
  1. Zhang Zhenyue
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  2. Du Keqin
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Correspondence to Zhang Zhenyue.

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Zhang, Z., Du, K. Successive projection method for solving the unbalanced Procrustes problem. SCI CHINA SER A 49, 971–986 (2006). https://doi.org/10.1007/s11425-006-0971-2

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  • DOI: https://doi.org/10.1007/s11425-006-0971-2

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