Partial Identifiability for Nonnegative Matrix Factorization · arXivDesk
2206.08022Jun 16, 202227 pages, 8 figures, 7 examples. This third version makes minor modifications. Paper accepted in SIAM J. on Matrix Analysis and Applications
Partial Identifiability for Nonnegative Matrix Factorization
Given a nonnegative matrix factorization, R, and a factorization rank, r, Exact nonnegative matrix factorization (Exact NMF) decomposes R as the product of two nonnegative matrices, C and S with r
Nearby in the stack
columns, such as
R=CS⊤
. A central research topic in the literature is the conditions under which such a decomposition is unique/identifiable, up to trivial ambiguities. In this paper, we focus on partial identifiability, that is, the uniqueness of a subset of columns of
C
and
S
. We start our investigations with the data-based uniqueness (DBU) theorem from the chemometrics literature. The DBU theorem analyzes all feasible solutions of Exact NMF, and relies on sparsity conditions on
C
and
S
. We provide a mathematically rigorous theorem of a recently published restricted version of the DBU theorem, relying only on simple sparsity and algebraic conditions: it applies to a particular solution of Exact NMF (as opposed to all feasible solutions) and allows us to guarantee the partial uniqueness of a single column of
C
or
S
. Second, based on a geometric interpretation of the restricted DBU theorem, we obtain a new partial identifiability result. This geometric interpretation also leads us to another partial identifiability result in the case
r=3
. Third, we show how partial identifiability results can be used sequentially to guarantee the identifiability of more columns of
C
and
S
. We illustrate these results on several examples, including one from the chemometrics literature.