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 



We thank anonymous referees for their detailed comments to improve the paper. This work is supported by the National Natural Science Foundation of China (Nos.11201426 and 11371365), the Zhejiang Provincial Natural Science Foundation of China (Nos.LQ12A01020, LQ13F030010, and LQ14G010004) and the Ministry of Education, Humanities and Social Sciences Research Project of China (No.13YJC910011).


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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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