# Robustness of Minimum-Volume Nonnegative Matrix Factorization under an Expanded Sufficiently Scattered Condition

arXiv: https://arxiv.org/abs/2511.04291 · HTML: https://mlsanity.com/paper/2511.04291/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 62 references that are in the map.
- First posted on arXiv 6 Nov 2025.

## Builds on (most cited first, 25 of 25)

- [Provably End-to-end Label-Noise Learning without Anchor Points](https://mlsanity.com/paper/2102.02400/) (2021)
- [Generalized Low Rank Models](https://mlsanity.com/paper/1410.0342/) (2014)
- [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)
- [Robust Volume Minimization-Based Matrix Factorization for Remote Sensing and Document Clustering](https://mlsanity.com/paper/1608.04290/) (2016)
- [Identifiability of the Simplex Volume Minimization Criterion for Blind Hyperspectral Unmixing: The No Pure-Pixel Case](https://mlsanity.com/paper/1406.5273/) (2014)
- [Crowdsourcing via Pairwise Co-occurrences: Identifiability and Algorithms](https://mlsanity.com/paper/1909.12325/) (2019)
- [Simplex-Structured Matrix Factorization: Sparsity-based Identifiability and Provably Correct Algorithms](https://mlsanity.com/paper/2007.11446/) (2020)
- [Maximum Volume Inscribed Ellipsoid: A New Simplex-Structured Matrix Factorization Framework via Facet Enumeration and Convex Optimization](https://mlsanity.com/paper/1708.02883/) (2017)
- [Blind Audio Source Separation with Minimum-Volume Beta-Divergence NMF](https://mlsanity.com/paper/1907.02404/) (2019)
- [Semidefinite Programming Based Preconditioning for More Robust Near-Separable Nonnegative Matrix Factorization](https://mlsanity.com/paper/1310.2273/) (2013)
- [Learning a Latent Simplex in Input-Sparsity Time](https://mlsanity.com/paper/2105.08005/) (2021)
- [Recovering Joint Probability of Discrete Random Variables from Pairwise Marginals](https://mlsanity.com/paper/2006.16912/) (2020)
- [Exact and Heuristic Algorithms for Semi-Nonnegative Matrix Factorization](https://mlsanity.com/paper/1410.7220/) (2014)
- [Probabilistic Simplex Component Analysis](https://mlsanity.com/paper/2103.10027/) (2021)
- [Ellipsoidal Rounding for Nonnegative Matrix Factorization Under Noisy Separability](https://mlsanity.com/paper/1309.5701/) (2013)
- [On Statistical Learning of Simplices: Unmixing Problem Revisited](https://mlsanity.com/paper/1810.07845/) (2018)
- [Deep Clustering with Incomplete Noisy Pairwise Annotations: A Geometric Regularization Approach](https://mlsanity.com/paper/2305.19391/) (2023)
- [Checking the Sufficiently Scattered Condition using a Global Non-Convex Optimization Software](https://mlsanity.com/paper/2402.06019/) (2024)
- [On the Robustness of the Successive Projection Algorithm](https://mlsanity.com/paper/2411.16195/) (2024)
- [Fixed points of the EM algorithm and nonnegative rank boundaries](https://mlsanity.com/paper/1312.5634/) (2013)
- [Robustness Analysis of Preconditioned Successive Projection Algorithm for General Form of Separable NMF Problem](https://mlsanity.com/paper/1506.08387/) (2015)
- [Minimum Volume Topic Modeling](https://mlsanity.com/paper/1904.02064/) (2019)
- [Identifiability of Nonnegative Tucker Decompositions -- Part I: Theory](https://mlsanity.com/paper/2505.12713/) (2025)
- [Fundamental Limits of Learning High-dimensional Simplices in Noisy Regimes](https://mlsanity.com/paper/2506.10101/) (2025)

## Abstract (arXiv)

Minimum-volume nonnegative matrix factorization (min-vol NMF) has been used successfully in many applications, such as hyperspectral imaging, chemical kinetics, spectroscopy, topic modeling, and audio source separation. However, its robustness to noise has been a long-standing open problem. In this paper, we prove that min-vol NMF identifies the groundtruth factors in the presence of noise under a condition referred to as the expanded sufficiently scattered condition which requires the data points to be sufficiently well scattered in the latent simplex generated by the basis vectors.