Bounded Simplex-Structured Matrix Factorization

Olivier Vu Thanh, Nicolas Gillis, Fabian Lecron

Abstract: 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 ≈ W H 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.