Multiplicative Updates for NMF with $β$-Divergences under Disjoint Equality Constraints
This work addresses a specific optimization challenge in NMF for researchers and practitioners in machine learning and signal processing, but it is incremental as it extends existing methods with constraints.
The paper tackles the problem of designing multiplicative updates for nonnegative matrix factorization with β-divergences under disjoint equality constraints, ensuring constraints are satisfied after each update and monotonic decrease in the objective function. It shows that the framework competes favorably with state-of-the-art methods on three NMF models, including those with sum-to-one and norm constraints.
Nonnegative matrix factorization (NMF) is the problem of approximating an input nonnegative matrix, $V$, as the product of two smaller nonnegative matrices, $W$ and $H$. In this paper, we introduce a general framework to design multiplicative updates (MU) for NMF based on $β$-divergences ($β$-NMF) with disjoint equality constraints, and with penalty terms in the objective function. By disjoint, we mean that each variable appears in at most one equality constraint. Our MU satisfy the set of constraints after each update of the variables during the optimization process, while guaranteeing that the objective function decreases monotonically. We showcase this framework on three NMF models, and show that it competes favorably the state of the art: (1)~$β$-NMF with sum-to-one constraints on the columns of $H$, (2) minimum-volume $β$-NMF with sum-to-one constraints on the columns of $W$, and (3) sparse $β$-NMF with $\ell_2$-norm constraints on the columns of $W$.