Nonnegative matrix factorization (NMF) is a linear dimensionality technique
for nonnegative data with applications such as image analysis, text mining,
audio source separation and hyperspectral unmixing. Given a data matrix M and
a factorization rank r, NMF looks for a nonnegative matrix W with r
columns and a nonnegative matrix H with r rows such that MβWH.
NMF is NP-hard to solve in general. However, it can be computed efficiently
under the separability assumption which requires that the basis vectors appear
as data points, that is, that there exists an index set K such that
W=M(:,K). In this paper, we generalize the separability
assumption: We only require that for each rank-one factor W(:,k)H(k,:) for
k=1,2,β¦,r, either W(:,k)=M(:,j) for some j or H(k,:)=M(i,:) for
some i. We refer to the corresponding problem as generalized separable NMF
(GS-NMF). We discuss some properties of GS-NMF and propose a convex
optimization model which we solve using a fast gradient method. We also propose
a heuristic algorithm inspired by the successive projection algorithm. To
verify the effectiveness of our methods, we compare them with several
state-of-the-art separable NMF algorithms on synthetic, document and image data
sets.Comment: 31 pages, 12 figures, 4 tables. We have added discussions about the
identifiability of the model, we have modified the first synthetic
experiment, we have clarified some aspects of the contributio