# Multiplicative Updates for NMF with β-Divergences under Disjoint Equality Constraints

arXiv: https://arxiv.org/abs/2010.16223 · HTML: https://mlsanity.com/paper/2010.16223/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Oct 2020.

## Built on by (most cited first, 8 of 8)

- [Bounded Simplex-Structured Matrix Factorization: Algorithms, Identifiability and Applications](https://mlsanity.com/paper/2209.12638/) (2022)
- [Majorization-minimization for Sparse Nonnegative Matrix Factorization with the β-divergence](https://mlsanity.com/paper/2207.06316/) (2022)
- [Partial Identifiability for Nonnegative Matrix Factorization](https://mlsanity.com/paper/2206.08022/) (2022)
- [A Second-Order Majorant Algorithm for Nonnegative Matrix Factorization](https://mlsanity.com/paper/2303.17992/) (2023)
- [Efficient Algorithms for Regularized Nonnegative Scale-invariant Low-rank Approximation Models](https://mlsanity.com/paper/2403.18517/) (2024)
- [Deep Nonnegative Matrix Factorization with Beta Divergences](https://mlsanity.com/paper/2309.08249/) (2023)
- [An Unobtrusive and Lightweight Ear-worn System for Continuous Epileptic Seizure Detection](https://mlsanity.com/paper/2401.05425/) (2024)
- [Block Majorization Minimization with Extrapolation and Application to β-NMF](https://mlsanity.com/paper/2401.06646/) (2024)

## Abstract (arXiv)

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 ℓ2-norm constraints on the columns of W.