Advertisement

Model Criticism for Regression on the Grassmannian

  • Yi Hong
  • Roland Kwitt
  • Marc Niethammer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9351)

Abstract

Reliable estimation of model parameters from data requires a suitable model. In this work, we investigate and extend a recent model criticism approach to evaluate regression models on the Grassmann manifold. Model criticism allows us to check if a model fits and if the underlying model assumptions are justified by the observed data. This is a critical step to check model validity which is often neglected in practice. Using synthetic data we demonstrate that the proposed model criticism approach can indeed reject models that are improper for observed data and that the approach can guide the model selection process. We study two real applications: degeneration of corpus callosum shapes during aging and developmental shape changes in the rat calvarium. Our experimental results suggest that the three tested regression models on the Grassmannian (equivalent to linear, time-warped, and cubic-spline regression in ℝ n , respectively) can all capture changes of the corpus callosum, but only the cubic-spline model is appropriate for shape changes of the rat calvarium. While our approach is developed for the Grassmannian, the principles are applicable to smooth manifolds in general.

Keywords

Regression Model Synthetic Data Reproduce Kernel Hilbert Space Model Criticism Statistical Model Criticism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Begelfor, E., Werman, W.: Affine invariance revisited. In: CVPR (2006)Google Scholar
  2. 2.
    Fletcher, P.T.: Geodesic regression and the theory of least squares on Riemannian manifolds. Int. J. Comput. Vision 105(2), 171–185 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gretton, A., Borgwardt, K.M., Rasch, M.J., Schölkopf, B., Smola, A.: A kernel two-sample test. J. Mach. Learn. Res. 13, 723–773 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Hamm, J., Lee, D.D.: Grassmann discriminant analysis: a unifying view on subspace-based learning. In: ICML (2008)Google Scholar
  5. 5.
    Harandi, M.T., Salzmann, M., Jayasumana, S., Hartley, R., Li, H.: Expanding the family of Grassmannian kernels: An embedding perspective. In: Fleet, D., Pajdla, T., Schiele, B., Tuytelaars, T. (eds.) ECCV 2014, Part VII. LNCS, vol. 8695, pp. 408–423. Springer, Heidelberg (2014)Google Scholar
  6. 6.
    Hong, Y., Kwitt, R., Singh, N., Davis, B., Vasconcelos, N., Niethammer, M.: Geodesic regression on the Grassmannian. In: Fleet, D., Pajdla, T., Schiele, B., Tuytelaars, T. (eds.) ECCV 2014, Part II. LNCS, vol. 8690, pp. 632–646. Springer, Heidelberg (2014)Google Scholar
  7. 7.
    Hong, Y., Singh, N., Kwitt, R., Vasconcelos, N., Niethammer, M.: Parametric regression on the Grassmannian. http://arxiv.org/abs/1505.03832
  8. 8.
    Lloyd, J.R., Ghahramani, Z.: Statistical model criticism using kernel two sample tests (Preprint). http://mlg.eng.cam.ac.uk/Lloyd/papers/
  9. 9.
    Niethammer, M., Huang, Y., Vialard, F.X.: Geodesic regression for image time-series. In: Fichtinger, G., Martel, A., Peters, T. (eds.) MICCAI, vol. 6892, pp. 655–662. Springer, Heidelberg (2011)Google Scholar
  10. 10.
    Wolf, L., Shashua, A.: Learning over sets using kernel principal angles. J. Mach. Learn. Res. 4, 913–931 (2003)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Zhang, M., Fletcher, P.T.: Probabilistic principal geodesic analysis. In: NIPS, pp. 1178–1186 (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Yi Hong
    • 1
  • Roland Kwitt
    • 3
  • Marc Niethammer
    • 1
    • 2
  1. 1.University of North Carolina (UNC) at Chapel HillChapel HillUS
  2. 2.Biomedical Research Imaging CenterUNC-Chapel HillChapel HillUS
  3. 3.Department of Computer ScienceUniversity of SalzburgSalzburgAustria

Personalised recommendations