The Equivalence Between Principal Component Analysis and Nearest Flat in the Least Square Sense



In this paper, we declare the equivalence between the principal component analysis and the nearest q-flat in the least square sense by showing that, for given m data points, the linear manifold with nearest distance is identical to the linear manifold with largest variance. Furthermore, from this observation, we give a new simpler proof for the approach to find the nearest q-flat.


Linear manifold Unsupervised learning Nearest q-flat Principal component analysis Eigenvalue decomposition 

Mathematics Subject Classification

15A18 58C40 


  1. 1.
    Bradley, P., Mangasarian, O.: k-plane clustering. J. Glob. Optim. 16(1), 23–32 (2000)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Tseng, P.: Nearest q-flat to m points. J. Optim. Theory Appl. 105(1), 249–252 (2000)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Zhang, T., Szlam, A., Wang, Y., Lerman, G.: Randomized hybrid linear modeling by local best-fit flats. In: 2010 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1927–1934, 2010Google Scholar
  4. 4.
    Chen, G., Lerman, G.: Spectral curvature clustering (SCC). Int. J. Comput. Vis. 81(3), 317–330 (2009)CrossRefGoogle Scholar
  5. 5.
    Chen, G., Lerman, G.: Foundations of a multi-way spectral clustering framework for hybrid linear modeling. Found. Comput. Math. 9(5), 517–558 (2009)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Wang, Y., Jiang, Y., Wu, Y., Zhou, Z.H.: Spectral clustering on multiple manifolds. IEEE Trans. Neural Netw. 22(7), 1149–1161 (2011)CrossRefGoogle Scholar
  7. 7.
    Shao, Y.H., Bai, L., Wang, Z., Hua, X.Y., Deng, N.Y.: Proximal plane clustering via eigenvalues. Proc. Comput. Sci. 17, 41–47 (2013)CrossRefGoogle Scholar
  8. 8.
    Amaldi, E., Dhyani, K., Liberti, L.: A two-phase heuristic for the bottleneck k-hyperplane clustering problem. Comput. Optim. Appl. 56(3), 619–633 (2013)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Szlam, A., Sapiro, G.: Discriminative k-metrics. In: 2009 ACM Conference on Proceedings of the International Conference on Machine Learning (ICML), pp. 1009–1016, 2009Google Scholar
  10. 10.
    Lerman, G., Zhang, T.: Robust recovery of multiple subspaces by geometric lp minimization. Ann. Stat. 39(5), 2686–2715 (2011)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Ramirez, I., Sprechmann, P., Sapiro, G.: Classification and clustering via dictionary learning with structured incoherence and shared features. In: 2010 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 3501–3508, 2010Google Scholar
  12. 12.
    Thiagarajan, J.J., Ramamurthy, K.N., Spanias, A.: Multilevel dictionary learning for sparse representation of images. In: 2011 IEEE Digital Signal Processing Workshop and IEEE Signal Processing Education Workshop (DSP/SPE), pp. 271–276, 2011Google Scholar
  13. 13.
    Abdi, H., Williams, L.J.: Principal component analysis. Wiley Interdiscip. Rev. 2(4), 433–459 (2010)CrossRefGoogle Scholar
  14. 14.
    Jolliffe, I.: Principal Component Analysis. Wiley, New York (2005)CrossRefGoogle Scholar
  15. 15.
    Ringnér, M.: What is principal component analysis? Nat. Biotechnol. 26(3), 303–304 (2008)CrossRefGoogle Scholar
  16. 16.
    Demšar, U., Harris, P., Brunsdon, C., Fotheringham, A.S., McLoone, S.: Principal component analysis on spatial data: an overview. Ann. Assoc. Am. Geogr. 103(1), 106–128 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Zhijiang CollegeZhejiang University of TechnologyHangzhouPeople’s Republic of China
  2. 2.College of ScienceChina Agricultural UniversityBeijingPeople’s Republic of China

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