# The Why and How of Nonnegative Matrix Factorization

arXiv: https://arxiv.org/abs/1401.5226 · HTML: https://mlsanity.com/paper/1401.5226/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jan 2014.

## Built on by (most cited first, 30 of 34)

- [Citizen Participation and Machine Learning for a Better Democracy](https://mlsanity.com/paper/2103.00508/) (2021)
- [Factorizer: A Scalable Interpretable Approach to Context Modeling for Medical Image Segmentation](https://mlsanity.com/paper/2202.12295/) (2022)
- [Probabilistic Simplex Component Analysis](https://mlsanity.com/paper/2103.10027/) (2021)
- [A Provable Splitting Approach for Symmetric Nonnegative Matrix Factorization](https://mlsanity.com/paper/2301.10499/) (2023)
- [Deep Clustering with Incomplete Noisy Pairwise Annotations: A Geometric Regularization Approach](https://mlsanity.com/paper/2305.19391/) (2023)
- [Sensing Cox Processes via Posterior Sampling and Positive Bases](https://mlsanity.com/paper/2110.11181/) (2021)
- [Analysis of Legal Documents via Non-negative Matrix Factorization Methods](https://mlsanity.com/paper/2104.14028/) (2021)
- [Identifiable Shared Component Analysis of Unpaired Multimodal Mixtures](https://mlsanity.com/paper/2409.19422/) (2024)
- [Non-negative matrix factorization algorithms generally improve topic model fits](https://mlsanity.com/paper/2105.13440/) (2021)
- [A Non-commutative Extension of Lee-Seung's Algorithm for Positive Semidefinite Factorizations](https://mlsanity.com/paper/2106.00293/) (2021)
- [A Fast Scale-Invariant Algorithm for Non-negative Least Squares with Non-negative Data](https://mlsanity.com/paper/2203.03808/) (2022)
- [Separable Quaternion Matrix Factorization for Polarization Images](https://mlsanity.com/paper/2207.14039/) (2022)
- [Concept-based Explanations using Non-negative Concept Activation Vectors and Decision Tree for CNN Models](https://mlsanity.com/paper/2211.10807/) (2022)
- [Privacy-Preserving Distributed Nonnegative Matrix Factorization](https://mlsanity.com/paper/2403.18326/) (2024)
- [Analysis of the robustness of NMF algorithms](https://mlsanity.com/paper/2106.02213/) (2021)
- [Supporting peace negotiations in the Yemen war through machine learning](https://mlsanity.com/paper/2207.11528/) (2022)
- [Multifidelity Covariance Estimation via Regression on the Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2307.12438/) (2023)
- [Stratified-NMF for Heterogeneous Data](https://mlsanity.com/paper/2311.10789/) (2023)
- [Non-Negative Reduced Biquaternion Matrix Factorization with Applications in Color Face Recognition](https://mlsanity.com/paper/2408.05582/) (2024)
- [GSVD-NMF: Recovering Missing Features in Non-negative Matrix Factorization](https://mlsanity.com/paper/2408.08260/) (2024)
- [Hybrid Ensemble Deep Graph Temporal Clustering for Spatiotemporal Data](https://mlsanity.com/paper/2409.12590/) (2024)
- [TACO: Training-free Sound Prompted Segmentation via Semantically Constrained Audio-visual CO-factorization](https://mlsanity.com/paper/2412.01488/) (2024)
- [Capturing Extreme Events in Turbulence using an Extreme Variational Autoencoder (xVAE)](https://mlsanity.com/paper/2502.04685/) (2025)
- [Bolster Hallucination Detection via Prompt-Guided Data Augmentation](https://mlsanity.com/paper/2510.15977/) (2025)
- [Time series forecasting from partial observations via Non-negative Matrix Factorization](https://mlsanity.com/paper/2102.05314/) (2021)
- [A Comprehensive Review on Non-Neural Networks Collaborative Filtering Recommendation Systems](https://mlsanity.com/paper/2106.10679/) (2021)
- [Multi-Channel Automatic Music Transcription Using Tensor Algebra](https://mlsanity.com/paper/2107.11250/) (2021)
- [Convex Latent Effect Logit Model via Sparse and Low-rank Decomposition](https://mlsanity.com/paper/2108.09859/) (2021)
- [Machine Learning for Mediation in Armed Conflicts](https://mlsanity.com/paper/2108.11942/) (2021)
- [Crime Hot-Spot Modeling via Topic Modeling and Relative Density Estimation](https://mlsanity.com/paper/2202.04176/) (2022)

## Abstract (arXiv)

Nonnegative matrix factorization (NMF) has become a widely used tool for the analysis of high-dimensional data as it automatically extracts sparse and meaningful features from a set of nonnegative data vectors. We first illustrate this property of NMF on three applications, in image processing, text mining and hyperspectral imaging --this is the why. Then we address the problem of solving NMF, which is NP-hard in general. We review some standard NMF algorithms, and also present a recent subclass of NMF problems, referred to as near-separable NMF, that can be solved efficiently (that is, in polynomial time), even in the presence of noise --this is the how. Finally, we briefly describe some problems in mathematics and computer science closely related to NMF via the nonnegative rank.