# Smoothed Separable Nonnegative Matrix Factorization

arXiv: https://arxiv.org/abs/2110.05528 · HTML: https://mlsanity.com/paper/2110.05528/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 42 references that are in the map.
- First posted on arXiv 11 Oct 2021.

## Builds on (most cited first, 12 of 12)

- [Hyperspectral Unmixing: Ground Truth Labeling, Datasets, Benchmark Performances and Survey](https://mlsanity.com/paper/1708.05125/) (2017)
- [Identifiability of the Simplex Volume Minimization Criterion for Blind Hyperspectral Unmixing: The No Pure-Pixel Case](https://mlsanity.com/paper/1406.5273/) (2014)
- [Simplex-Structured Matrix Factorization: Sparsity-based Identifiability and Provably Correct Algorithms](https://mlsanity.com/paper/2007.11446/) (2020)
- [Successive Nonnegative Projection Algorithm for Robust Nonnegative Blind Source Separation](https://mlsanity.com/paper/1310.7529/) (2013)
- [Maximum Volume Inscribed Ellipsoid: A New Simplex-Structured Matrix Factorization Framework via Facet Enumeration and Convex Optimization](https://mlsanity.com/paper/1708.02883/) (2017)
- [Robust Near-Separable Nonnegative Matrix Factorization Using Linear Optimization](https://mlsanity.com/paper/1302.4385/) (2013)
- [Learning a Latent Simplex in Input-Sparsity Time](https://mlsanity.com/paper/2105.08005/) (2021)
- [Column subset selection is NP-complete](https://mlsanity.com/paper/1701.02764/) (2017)
- [Probabilistic Simplex Component Analysis](https://mlsanity.com/paper/2103.10027/) (2021)
- [Finding a latent k-simplex in O(k . nnz(data)) time via Subset Smoothing](https://mlsanity.com/paper/1904.06738/) (2019)
- [Intersecting Faces: Non-negative Matrix Factorization With New Guarantees](https://mlsanity.com/paper/1507.02189/) (2015)
- [A Fast Hyperplane-Based Minimum-Volume Enclosing Simplex Algorithm for Blind Hyperspectral Unmixing](https://mlsanity.com/paper/1510.08917/) (2015)

## Built on by (most cited first, 5 of 5)

- [Improved Algorithm and Bounds for Successive Projection](https://mlsanity.com/paper/2403.11013/) (2024)
- [On the Robustness of the Successive Projection Algorithm](https://mlsanity.com/paper/2411.16195/) (2024)
- [Matrix Factorization Framework for Community Detection under the Degree-Corrected Block Model](https://mlsanity.com/paper/2601.06262/) (2026)
- [A Provably-Correct and Robust Convex Model for Smooth Separable NMF](https://mlsanity.com/paper/2511.07109/) (2025)
- [Unmixing ATR-μFTIR spectroscopic images of cross-sections of historical oil paintings](https://mlsanity.com/paper/2603.06673/) (2026)

## Abstract (arXiv)

Given a set of data points belonging to the convex hull of a set of vertices, a key problem in linear algebra, signal processing, data analysis and machine learning is to estimate these vertices in the presence of noise. Many algorithms have been developed under the assumption that there is at least one nearby data point to each vertex; two of the most widely used ones are vertex component analysis (VCA) and the successive projection algorithm (SPA). This assumption is known as the pure-pixel assumption in blind hyperspectral unmixing, and as the separability assumption in nonnegative matrix factorization. More recently, Bhattacharyya and Kannan (ACM-SIAM Symposium on Discrete Algorithms, 2020) proposed an algorithm for learning a latent simplex (ALLS) that relies on the assumption that there is more than one nearby data point to each vertex. In that scenario, ALLS is probalistically more robust to noise than algorithms based on the separability assumption. In this paper, inspired by ALLS, we propose smoothed VCA (SVCA) and smoothed SPA (SSPA) that generalize VCA and SPA by assuming the presence of several nearby data points to each vertex. We illustrate the effectiveness of SVCA and SSPA over VCA, SPA and ALLS on synthetic data sets, on the unmixing of hyperspectral images, and on feature extraction on facial images data sets. In addition, our study highlights new theoretical results for VCA.