Bounded Simplex-Structured Matrix Factorization: Algorithms, Identifiability and Applications

In this article, we propose a new low-rank matrix factorization model dubbed bounded simplex-structured matrix factorization (BSSMF). Given an input matrix
$X$
and a factorization rank
$r$
, BSSMF looks for a matrix
$W$
with
$r$
columns and a matrix
$H$
with
$r$
rows such that
$X \approx WH$
where the entries in each column of
$W$
are bounded, that is, they belong to given intervals, and the columns of
$H$
belong to the probability simplex, that is,
$H$
is column stochastic. BSSMF generalizes nonnegative matrix factorization (NMF), and simplex-structured matrix factorization (SSMF). BSSMF is particularly well suited when the entries of the input matrix
$X$
belong to a given interval; for example when the rows of
$X$
represent images, or
$X$
is a rating matrix such as in the Netflix and MovieLens datasets where the entries of
$X$
belong to the interval [1,5]. The simplex-structured matrix
$H$
not only leads to an easily understandable decomposition providing a soft clustering of the columns of
$X$
, but implies that the entries of each column of
$WH$
belong to the same intervals as the columns of
$W$
. In this article, we first propose a fast algorithm for BSSMF, even in the presence of missing data in
$X$
. Then we provide identifiability conditions for BSSMF, that is, we provide conditions under which BSSMF admits a unique decomposition, up to trivial ambiguities. Finally, we illustrate the effectiveness of BSSMF on two applications: extraction of features in a set of images, and the matrix completion problem for recommender systems.