# Majorization-minimization for Sparse Nonnegative Matrix Factorization with the β-divergence

arXiv: https://arxiv.org/abs/2207.06316 · HTML: https://mlsanity.com/paper/2207.06316/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 49 references that are in the map.
- First posted on arXiv 13 Jul 2022.

## Builds on (most cited first, 7 of 7)

- [Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applications](https://mlsanity.com/paper/1803.01257/) (2018)
- [On Identifiability of Nonnegative Matrix Factorization](https://mlsanity.com/paper/1709.00614/) (2017)
- [Nonlinear hyperspectral unmixing with robust nonnegative matrix factorization](https://mlsanity.com/paper/1401.5649/) (2014)
- [Ordinal Non-negative Matrix Factorization for Recommendation](https://mlsanity.com/paper/2006.01034/) (2020)
- [Multiplicative Updates for NMF with β-Divergences under Disjoint Equality Constraints](https://mlsanity.com/paper/2010.16223/) (2020)
- [A Unified Convergence Analysis of the Multiplicative Update Algorithm for Regularized Nonnegative Matrix Factorization](https://mlsanity.com/paper/1609.00951/) (2016)
- [Log-based Sparse Nonnegative Matrix Factorization for Data Representation](https://mlsanity.com/paper/2204.10647/) (2022)

## Built on by (most cited first, 4 of 4)

- [An Accelerated Alternating Partial Bregman Algorithm for ReLU-based Matrix Decomposition](https://mlsanity.com/paper/2503.02386/) (2025)
- [Efficient Algorithms for Regularized Nonnegative Scale-invariant Low-rank Approximation Models](https://mlsanity.com/paper/2403.18517/) (2024)
- [Block Majorization Minimization with Extrapolation and Application to β-NMF](https://mlsanity.com/paper/2401.06646/) (2024)
- [On the Connection Between Non-negative Matrix Factorization and Latent Dirichlet Allocation](https://mlsanity.com/paper/2405.20542/) (2024)

## Abstract (arXiv)

This article introduces new multiplicative updates for nonnegative matrix factorization with the β-divergence and sparse regularization of one of the two factors (say, the activation matrix). It is well known that the norm of the other factor (the dictionary matrix) needs to be controlled in order to avoid an ill-posed formulation. Standard practice consists in constraining the columns of the dictionary to have unit norm, which leads to a nontrivial optimization problem. Our approach leverages a reparametrization of the original problem into the optimization of an equivalent scale-invariant objective function. From there, we derive block-descent majorization-minimization algorithms that result in simple multiplicative updates for either ℓ1-regularization or the more "aggressive" log-regularization. In contrast with other state-of-the-art methods, our algorithms are universal in the sense that they can be applied to any β-divergence (i.e., any value of β) and that they come with convergence guarantees. We report numerical comparisons with existing heuristic and Lagrangian methods using various datasets: face images, an audio spectrogram, hyperspectral data, and song play counts. We show that our methods obtain solutions of similar quality at convergence (similar objective values) but with significantly reduced CPU times.