Pre-Scaling Anisotropic Orthogonal Procrustes Analysis Based on Gradient Descent over Matrix Manifold

  • Peng ZhangEmail author
  • Zhou Sun
  • Chunbo Fan
  • Yi Ding
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9141)


This paper proposes a pre-scaling extension of the Orthogonal Procrustes Analysis (OPA), where anisotropic scaling occurs before rigid motion. We propose an efficient algorithm to solve this problem based on gradient descent method over matrix manifold. We show that the proposed algorithm is monotonically convergent and provide an acceleration procedure. Its performance is validated through a series of numerical simulations.


Procrustes analysis Gradient descent Stiefel manifold Pre-scaling 


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  1. 1.
    Hurley, J.R., Cattell, R.B.: The Procrustes program: producing direct rotation to test a hypothesized factor structure. Behavioral Science 7, 258–262 (1962)CrossRefGoogle Scholar
  2. 2.
    Schonemann, P.H., Carroll, R.M.: Fitting one matrix to another under choice of a central dilation and a rigid motion. Psychometrika 35, 245–255 (1970)CrossRefGoogle Scholar
  3. 3.
    Gower, J.C.: Statistical methods of comparing different multivariate analyses of the same data. In: Mathematics in the Archeological and Historical Sciences, pp. 138–149. University Press (1971)Google Scholar
  4. 4.
    Ten Berge, J.M.F.: Orthogonal Procrustes rotation for two or more matrices. Psychometrika 42, 267–276 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Goodall, C.: Procrustes methods in the statistical analysis of shape. Journal of the Royal Statistical Society, Series B 53, 285–339 (1991)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Dryden, I., Mardia, K.: Statistical shape analysis. John Wiley and Sons (1998)Google Scholar
  7. 7.
    Goldberg, Y., Ritov, Y.: Local procrustes for manifold embedding: a measure of embedding quality and embedding algorithms. Machine Learning 77(1), 1–25 (2009)CrossRefGoogle Scholar
  8. 8.
    Garro, V., Crosilla, F., Fusiello, A.: Solving the PnP problem with anisotropic orthogonal Procrustes analysis. In: Proceedings of the 2012 Second Joint 3DIM/3DPVT Conference: 3D Imaging, Modeling, Processing, Visualization & Transmission, 262–269 (2012)Google Scholar
  9. 9.
    Chen, E.C., McLeod, A.J., Jayarathne, U.L., Peter, T.M.: Solving for free-hand and real-time 3d ultrasound calibration with anisotropic orthogonal Procrustes analysis. In: Proceedings of SPIE, vol. 9036, 90361Z-1-7 (2014)Google Scholar
  10. 10.
    Gower, J.C.: Dijksterhuis, G.B.: Procrustes problems. Oxford University Press (2004)Google Scholar
  11. 11.
    Dosse, M.B.: Extension of Generalized Procrustes Analysis. Agrostat, Rennes, pp. 1–16 (2004)Google Scholar
  12. 12.
    Dosse, M.B., Berge, J.T.: Anisotropic orthogonal Procrustes Analysis. Journal of Classification 27, 111–128 (2010)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Zhang, P., Ren, Y., Zhang, B.: A new embedding quality assessment method for manifold learning. Neurocomputing 97, 251–266 (2012)CrossRefGoogle Scholar
  14. 14.
    Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton, NJ, USA (2007)Google Scholar
  15. 15.
    Magnus, J.R., Neudecker, H.: Matrix differential calculus with applications in statistics and econometrics. 2nd edn. John Wiley and Sons (1999)Google Scholar
  16. 16.
    Tenenbaum, J.B., Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)CrossRefGoogle Scholar
  17. 17.
    Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500), 2323–2326 (2000)CrossRefGoogle Scholar
  18. 18.
    Zhang, Z., Zha, H.: Principal manifolds and nonlinear dimensionality reduction via tangent space alignment. SIAM Journal on Scientific Computing 26(1), 313–338 (2005)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Valdez, F., Melin, P., Castillo, O.: An improved evolutionary method with fuzzy logic for combining Particle Swarm Optimization and Genetic Algorithms. Applied Soft Computing 11(2), 2625–2632 (2010)CrossRefGoogle Scholar
  20. 20.
    Precup, R.-E., David, R.-C., Petriu, E.M., Preitl, S., Paul, A.S.: Gravitational Search Algorithm-Based Tuning of Fuzzy Control Systems with a Reduced Parametric Sensitivity. In: Gaspar-Cunha, A., Takahashi, R., Schaefer, G., Costa, L. (eds.) Soft Computing in Industrial Applications. AISC, vol. 96, pp. 141–150. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  21. 21.
    Wu, Z., Chow, T., Cheng, S., Shi, Y.: Contour gradient optimization. International Journal of Swarm Intelligence Research 4(2), 1–28 (2013)CrossRefGoogle Scholar
  22. 22.
    El-Hefnawy, N.: Solving bi-level problems using modified particle swarm optimization algorithm. International Journal of Artificial Intelligence 12(2), 88–101 (2014)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Data Center, National Disaster Reduction Center of ChinaBeijingPeople’s Republic of China

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