# Dual Simplex Volume Maximization for Simplex-Structured Matrix Factorization

arXiv: https://arxiv.org/abs/2403.20197 · HTML: https://mlsanity.com/paper/2403.20197/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 34 references that are in the map.
- First posted on arXiv 29 Mar 2024.

## Builds on (most cited first, 13 of 13)

- [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)
- [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)
- [Computing Large-Scale Matrix and Tensor Decomposition with Structured Factors: A Unified Nonconvex Optimization Perspective](https://mlsanity.com/paper/2006.08183/) (2020)
- [Polytopic Matrix Factorization: Determinant Maximization Based Criterion and Identifiability](https://mlsanity.com/paper/2202.09638/) (2022)
- [Learning a Latent Simplex in Input-Sparsity Time](https://mlsanity.com/paper/2105.08005/) (2021)
- [On the Complexity of Robust PCA and ℓ1-norm Low-Rank Matrix Approximation](https://mlsanity.com/paper/1509.09236/) (2015)
- [Exact and Heuristic Algorithms for Semi-Nonnegative Matrix Factorization](https://mlsanity.com/paper/1410.7220/) (2014)
- [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, 4 of 4)

- [Adaptive Multi-Order Graph Regularized NMF with Dual Sparsity for Hyperspectral Unmixing](https://mlsanity.com/paper/2503.19258/) (2025)
- [Nonnegative matrix factorizations and related compositional models: Equivalence, identifiability, and an application on the grain-size analysis of sediments](https://mlsanity.com/paper/2512.22282/) (2025)
- [Maximum-Volume Nonnegative Matrix Factorization](https://mlsanity.com/paper/2602.04795/) (2026)
- [Identification of NMF by choosing maximum-volume basis vectors](https://mlsanity.com/paper/2603.24227/) (2026)

## Abstract (arXiv)

Simplex-structured matrix factorization (SSMF) is a generalization of nonnegative matrix factorization, a fundamental interpretable data analysis model, and has applications in hyperspectral unmixing and topic modeling. To obtain identifiable solutions, a standard approach is to find minimum-volume solutions. By taking advantage of the duality/polarity concept for polytopes, we convert minimum-volume SSMF in the primal space to a maximum-volume problem in the dual space. We first prove the identifiability of this maximum-volume dual problem. Then, we use this dual formulation to provide a novel optimization approach which bridges the gap between two existing families of algorithms for SSMF, namely volume minimization and facet identification. Numerical experiments show that the proposed approach performs favorably compared to the state-of-the-art SSMF algorithms.