BOUNDED SIMPLEX-STRUCTURED MATRIX FACTORIZATION

In this paper, 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 unit 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 $X$ themselves belong to a given interval; for example when the columns of $X$ represent images. In this paper, we first provide identifiability conditions for BSSMF, that is, we provide conditions under which BSSMF admits a unique decomposition, up to trivial ambiguities. Then we propose a fast inertial algorithm for BSSMF. Finally, we illustrate the effectiveness of BSSMF to obtain interpretable features in the MNIST dataset.