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

TL;DR

This paper introduces a flexible multiplicative update framework for $eta$-divergence-based NMF with disjoint equality constraints, ensuring monotonic decrease of the objective and applicability to various models.

Contribution

It presents a novel MU framework for $eta$-NMF with disjoint equality constraints, guaranteeing constraint satisfaction and objective decrease during optimization.

Findings

Framework performs favorably against state-of-the-art methods.

Ensures constraints are satisfied after each update.

Applicable to multiple NMF models with different constraints.

Abstract

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 $\beta$-divergences ($\beta$-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)~$\beta$-NMF with sum-to-one constraints on the columns of $H$, (2) minimum-volume $\beta$-NMF with sum-to-one constraints on the columns of $W$, and (3) sparse $\beta$-NMF with $\ell_2$-norm constraints on the columns of $W$.