When is non-negative matrix decomposition unique?

Scott Rickard, Andrzej Cichocki

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

10 Citations (SciVal)

Abstract

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.

Original languageEnglish
Title of host publicationCISS 2008, The 42nd Annual Conference on Information Sciences and Systems
Pages1091-1092
Number of pages2
DOIs
Publication statusPublished - 2008
Externally publishedYes
EventCISS 2008, 42nd Annual Conference on Information Sciences and Systems - Princeton, NJ, United States
Duration: 19 Mar 200821 Mar 2008

Publication series

NameCISS 2008, The 42nd Annual Conference on Information Sciences and Systems

Conference

ConferenceCISS 2008, 42nd Annual Conference on Information Sciences and Systems
Country/TerritoryUnited States
CityPrinceton, NJ
Period19/03/0821/03/08

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