Editing Principal component analysis (section)

=== {{math|<var>K</var>}}-means clustering ===
It has been asserted that the relaxed solution of [[k-means clustering|{{math|<var>k</var>}}-means clustering]], specified by the cluster indicators, is given by the principal components, and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace.<ref>{{cite journal|author=H. Zha |author2=C. Ding |author3=M. Gu |author4=X. He |author5=H.D. Simon|title=Spectral Relaxation for K-means Clustering|journal=Neural Information Processing Systems Vol.14 (NIPS 2001)|pages=1057–1064|date=Dec 2001|url=http://ranger.uta.edu/~chqding/papers/Zha-Kmeans.pdf}}</ref><ref>{{cite journal|author=Chris Ding |author2=Xiaofeng He|title=K-means Clustering via Principal Component Analysis|journal=Proc. Of Int'l Conf. Machine Learning (ICML 2004)|pages=225–232|date=July 2004|url=http://ranger.uta.edu/~chqding/papers/KmeansPCA1.pdf}}</ref> However, that PCA is a useful relaxation of {{math|<var>k</var>}}-means clustering was not a new result,<ref>{{cite journal | title = Clustering large graphs via the singular value decomposition | journal = Machine Learning | year = 2004 | first = P. | last = Drineas |author2=A. Frieze |author3=R. Kannan |author4=S. Vempala |author5=V. Vinay | volume = 56 | issue = 1–3 | pages = 9–33| url = http://www.cc.gatech.edu/~vempala/papers/dfkvv.pdf | access-date = 2012-08-02 | doi=10.1023/b:mach.0000033113.59016.96| s2cid = 5892850 | doi-access = free }}</ref> and it is straightforward to uncover counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions.<ref>{{cite book | title = Dimensionality reduction for k-means clustering and low rank approximation (Appendix B) | year = 2014 | first = M. | last = Cohen |author2=S. Elder |author3=C. Musco |author4=C. Musco |author5=M. Persu | arxiv = 1410.6801|bibcode=2014arXiv1410.6801C}}</ref>