Journal of Optimization Theory and Applications

, Volume 166, Issue 1, pp 278–284

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


DOI: 10.1007/s10957-014-0647-y

Cite this article as:
Shao, YH. & Deng, NY. J Optim Theory Appl (2015) 166: 278. doi:10.1007/s10957-014-0647-y


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 

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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