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Journal Articles SIAM Journal on Matrix Analysis and Applications Year : 2021

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

Valentin Leplat
• Function : Author
Nicolas Gillis
• Function : Author
Jérôme Idier

#### 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 $W$.

### Dates and versions

hal-02993314 , version 1 (06-11-2020)

### Identifiers

• HAL Id : hal-02993314 , version 1
• ARXIV :
• DOI :

### Cite

Valentin Leplat, Nicolas Gillis, Jérôme Idier. Multiplicative Updates for NMF with $\beta$-Divergences under Disjoint Equality Constraints. SIAM Journal on Matrix Analysis and Applications, 2021, 42 (2), pp.730-752. ⟨10.1137/20M1377278⟩. ⟨hal-02993314⟩

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