Editing Principal component analysis (section)

=== Network component analysis ===
Given a matrix <math>E</math>, it tries to decompose it into two matrices such that <math>E=AP
</math>. A key difference from techniques such as PCA and ICA is that some of the entries of <math>A</math> are constrained to be 0. Here <math>P</math> is termed the regulatory layer. While in general such a decomposition can have multiple solutions, they prove that if the following conditions are satisfied :
# <math>A</math> has full column rank
# Each column of <math>A</math> must have at least <math>L-1</math> zeroes where <math>L</math> is the number of columns of <math>A</math> (or alternatively the number of rows of <math>P</math>). The justification for this criterion is that if a node is removed from the regulatory layer along with all the output nodes connected to it, the result must still be characterized by a connectivity matrix with full column rank.
# <math>P</math> must have full row rank.
then the decomposition is unique up to multiplication by a scalar.<ref>{{Cite journal|title = Network component analysis: Reconstruction of regulatory signals in biological systems|last1=Liao|first1=J. C.|last2=Boscolo|first2=R.|last3=Yang|first3=Y.-L.|last4=Tran|first4=L. M.|last5=Sabatti|first5=C.|author5-link=Chiara Sabatti|last6=Roychowdhury|first6=V. P.|journal=Proceedings of the National Academy of Sciences|volume=100|issue=26|date=2003|pages=15522–15527|doi=10.1073/pnas.2136632100|pmid = 14673099|pmc = 307600|bibcode = 2003PNAS..10015522L|doi-access=free}}</ref>