Multiplicative Updates for NMF with $\beta$-Divergences under Disjoint Equality Constraints

Nonnegative matrix factorization (NMF) is the problem of approximating an input nonnegative matrix, $V$, as product two smaller matrices, $W$ and $H$. In this paper, we introduce a general framework to design multiplicative updates (MU) for NMF based on $\beta$-divergences ($\beta$-NMF) with disjoint equality constraints, penalty terms in objective function. By disjoint, mean that each variable appears at most one constraint. Our MU satisfy set constraints after update variables during optimization process, while guaranteeing function decreases monotonically. We showcase three models, show it competes favorably state art: (1)~$\beta$-NMF sum-to-one columns $H$, (2) minimum-volume $\beta$-NMF $W$, (3) sparse $\ell_2$-norm $W$.