  Nonnegative matrix factorization (NMF) is a popular method in machine
learning and signal processing to decompose a given nonnegative matrix into two
nonnegative matrices. In this paper, to solve NMF, we propose new algorithms,
called majorization-minimization Bregman proximal gradient algorithm (MMBPG)
and MMBPG with extrapolation (MMBPGe). MMBPG and MMBPGe minimize an auxiliary
function majorizing the Kullback--Leibler (KL) divergence loss by the existing
Bregman proximal gradient algorithms. While existing KL-based NMF methods
update each variable alternately, proposed algorithms update all variables
simultaneously. The proposed MMBPG and MMBPGe are equipped with a separable
Bregman distance that satisfies the smooth adaptable property and that makes
its subproblem solvable in closed forms. We also proved that even though these
algorithms are designed to minimize an auxiliary function, MMBPG and MMBPGe
monotonically decrease the objective function and a potential function,
respectively. Using this fact, we establish that a sequence generated by
MMBPG(e) globally converges to a Karush--Kuhn--Tucker (KKT) point. In numerical
experiments, we compared proposed algorithms with existing algorithms on
synthetic data.
