TY - GEN

T1 - When is non-negative matrix decomposition unique?

AU - Rickard, Scott

AU - Cichocki, Andrzej

PY - 2008

Y1 - 2008

N2 - In this paper, we discuss why non-negative matrix factorization (NMF) potentially works for zero-grounded non-negative components and why it fails when the components are not zero-grounded. We show the demixing process is not uniquely defined (up to the usual permutation/scaling ambiguity) when the original matrices are not zero-grounded. If fact, zero-groundedness alone is not enough. The key observation is that if each component has at least one point for which it is the only active component, the solution is unique. When the non-negative matrices are not zero-grounded, no such point exists and the solution space contains demixtures which are linear combinations of the original components. Thus, the NMF problem has a unique solution for matrices with disjoint components, a condition we call Subset Monomial Disjoint (SMD). The SMD condition is sufficient, but not necessary for NMF to have a unique decomposition, whereas the zero-grounded condition is necessary, but not sufficient.

AB - In this paper, we discuss why non-negative matrix factorization (NMF) potentially works for zero-grounded non-negative components and why it fails when the components are not zero-grounded. We show the demixing process is not uniquely defined (up to the usual permutation/scaling ambiguity) when the original matrices are not zero-grounded. If fact, zero-groundedness alone is not enough. The key observation is that if each component has at least one point for which it is the only active component, the solution is unique. When the non-negative matrices are not zero-grounded, no such point exists and the solution space contains demixtures which are linear combinations of the original components. Thus, the NMF problem has a unique solution for matrices with disjoint components, a condition we call Subset Monomial Disjoint (SMD). The SMD condition is sufficient, but not necessary for NMF to have a unique decomposition, whereas the zero-grounded condition is necessary, but not sufficient.

UR - http://www.scopus.com/inward/record.url?scp=51849087384&partnerID=8YFLogxK

U2 - 10.1109/CISS.2008.4558681

DO - 10.1109/CISS.2008.4558681

M3 - Conference contribution

AN - SCOPUS:51849087384

SN - 9781424422470

T3 - CISS 2008, The 42nd Annual Conference on Information Sciences and Systems

SP - 1091

EP - 1092

BT - CISS 2008, The 42nd Annual Conference on Information Sciences and Systems

T2 - CISS 2008, 42nd Annual Conference on Information Sciences and Systems

Y2 - 19 March 2008 through 21 March 2008

ER -