# Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applications

arXiv: https://arxiv.org/abs/1803.01257 · HTML: https://mlsanity.com/paper/1803.01257/

## Key facts

- Cited by 55 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Mar 2018.

## Built on by (most cited first, 30 of 55)

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- [Deep Spectrum Cartography: Completing Radio Map Tensors Using Learned Neural Models](https://mlsanity.com/paper/2105.00177/) (2021)
- [Polytopic Matrix Factorization: Determinant Maximization Based Criterion and Identifiability](https://mlsanity.com/paper/2202.09638/) (2022)
- [FactorizePhys: Matrix Factorization for Multidimensional Attention in Remote Physiological Sensing](https://mlsanity.com/paper/2411.01542/) (2024)
- [Deep Learning From Crowdsourced Labels: Coupled Cross-entropy Minimization, Identifiability, and Regularization](https://mlsanity.com/paper/2306.03288/) (2023)
- [Graph-based Neural Acceleration for Nonnegative Matrix Factorization](https://mlsanity.com/paper/2202.00264/) (2022)
- [A Unified Framework for Optimization-Based Graph Coarsening](https://mlsanity.com/paper/2210.00437/) (2022)
- [Probabilistic Simplex Component Analysis](https://mlsanity.com/paper/2103.10027/) (2021)
- [Identifiability-Guaranteed Simplex-Structured Post-Nonlinear Mixture Learning via Autoencoder](https://mlsanity.com/paper/2106.09070/) (2021)
- [Joint Majorization-Minimization for Nonnegative Matrix Factorization with the β-divergence](https://mlsanity.com/paper/2106.15214/) (2021)
- [Crowdsourcing via Annotator Co-occurrence Imputation and Provable Symmetric Nonnegative Matrix Factorization](https://mlsanity.com/paper/2106.07193/) (2021)
- [Memory-Efficient Convex Optimization for Self-Dictionary Separable Nonnegative Matrix Factorization: A Frank-Wolfe Approach](https://mlsanity.com/paper/2109.11135/) (2021)
- [Fast and Structured Block-Term Tensor Decomposition For Hyperspectral Unmixing](https://mlsanity.com/paper/2205.03798/) (2022)
- [Biologically-Plausible Determinant Maximization Neural Networks for Blind Separation of Correlated Sources](https://mlsanity.com/paper/2209.12894/) (2022)
- [Bounded Simplex-Structured Matrix Factorization: Algorithms, Identifiability and Applications](https://mlsanity.com/paper/2209.12638/) (2022)
- [Deep Clustering with Incomplete Noisy Pairwise Annotations: A Geometric Regularization Approach](https://mlsanity.com/paper/2305.19391/) (2023)
- [Towards Identifiable Unsupervised Domain Translation: A Diversified Distribution Matching Approach](https://mlsanity.com/paper/2401.09671/) (2024)
- [Majorization-minimization for Sparse Nonnegative Matrix Factorization with the β-divergence](https://mlsanity.com/paper/2207.06316/) (2022)
- [Dual Simplex Volume Maximization for Simplex-Structured Matrix Factorization](https://mlsanity.com/paper/2403.20197/) (2024)
- [Diverse Influence Component Analysis: A Geometric Approach to Nonlinear Mixture Identifiability](https://mlsanity.com/paper/2510.17040/) (2025)
- [Multiple Kernel Representation Learning on Networks](https://mlsanity.com/paper/2106.05057/) (2021)
- [Refinement of Hottopixx Method for Nonnegative Matrix Factorization Under Noisy Separability](https://mlsanity.com/paper/2109.02863/) (2021)
- [Partial Identifiability for Nonnegative Matrix Factorization](https://mlsanity.com/paper/2206.08022/) (2022)
- [A consistent and flexible framework for deep matrix factorizations](https://mlsanity.com/paper/2206.10693/) (2022)
- [Correlative Information Maximization Based Biologically Plausible Neural Networks for Correlated Source Separation](https://mlsanity.com/paper/2210.04222/) (2022)
- [Checking the Sufficiently Scattered Condition using a Global Non-Convex Optimization Software](https://mlsanity.com/paper/2402.06019/) (2024)
- [Learning From Crowdsourced Noisy Labels: A Signal Processing Perspective](https://mlsanity.com/paper/2407.06902/) (2024)
- [Identifiable Shared Component Analysis of Unpaired Multimodal Mixtures](https://mlsanity.com/paper/2409.19422/) (2024)
- [Structured Sparse Non-negative Matrix Factorization with L20-Norm for scRNA-seq Data Analysis](https://mlsanity.com/paper/2104.13171/) (2021)

## Abstract (arXiv)

Nonnegative matrix factorization (NMF) has become a workhorse for signal and data analytics, triggered by its model parsimony and interpretability. Perhaps a bit surprisingly, the understanding to its model identifiability---the major reason behind the interpretability in many applications such as topic mining and hyperspectral imaging---had been rather limited until recent years. Beginning from the 2010s, the identifiability research of NMF has progressed considerably: Many interesting and important results have been discovered by the signal processing (SP) and machine learning (ML) communities. NMF identifiability has a great impact on many aspects in practice, such as ill-posed formulation avoidance and performance-guaranteed algorithm design. On the other hand, there is no tutorial paper that introduces NMF from an identifiability viewpoint. In this paper, we aim at filling this gap by offering a comprehensive and deep tutorial on model identifiability of NMF as well as the connections to algorithms and applications. This tutorial will help researchers and graduate students grasp the essence and insights of NMF, thereby avoiding typical `pitfalls' that are often times due to unidentifiable NMF formulations. This paper will also help practitioners pick/design suitable factorization tools for their own problems.