# Simplex-Structured Matrix Factorization: Sparsity-based Identifiability and Provably Correct Algorithms

arXiv: https://arxiv.org/abs/2007.11446 · HTML: https://mlsanity.com/paper/2007.11446/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jul 2020.

## Built on by (most cited first, 21 of 21)

- [Smoothed Separable Nonnegative Matrix Factorization](https://mlsanity.com/paper/2110.05528/) (2021)
- [Bounded Simplex-Structured Matrix Factorization: Algorithms, Identifiability and Applications](https://mlsanity.com/paper/2209.12638/) (2022)
- [Quadratic Matrix Factorization with Applications to Manifold Learning](https://mlsanity.com/paper/2301.12965/) (2023)
- [Dual Simplex Volume Maximization for Simplex-Structured Matrix Factorization](https://mlsanity.com/paper/2403.20197/) (2024)
- [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)
- [On the Robustness of the Successive Projection Algorithm](https://mlsanity.com/paper/2411.16195/) (2024)
- [Identifiability of Nonnegative Tucker Decompositions -- Part I: Theory](https://mlsanity.com/paper/2505.12713/) (2025)
- [Identifiability in Two-Layer Sparse Matrix Factorization](https://mlsanity.com/paper/2110.01235/) (2021)
- [Deep Nonnegative Matrix Factorization with Beta Divergences](https://mlsanity.com/paper/2309.08249/) (2023)
- [Subspace-Constrained Quadratic Matrix Factorization: Algorithm and Applications](https://mlsanity.com/paper/2411.04717/) (2024)
- [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)
- [Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions](https://mlsanity.com/paper/2605.24113/) (2026)
- [Robust Matrix Factorization with Grouping Effect](https://mlsanity.com/paper/2106.13681/) (2021)
- [Regularized Projection Matrix Approximation with Applications to Community Detection](https://mlsanity.com/paper/2405.16598/) (2024)
- [Robustness of Minimum-Volume Nonnegative Matrix Factorization under an Expanded Sufficiently Scattered Condition](https://mlsanity.com/paper/2511.04291/) (2025)
- [A Provably-Correct and Robust Convex Model for Smooth Separable NMF](https://mlsanity.com/paper/2511.07109/) (2025)
- [Regularized Optimization on Grassmann Manifold: Theory, Algorithm and Applications](https://mlsanity.com/paper/2607.21039/) (2026)
- [Smooth Reparameterizations of Functions on Simplicial Product Spaces: Applications to Probabilistic Tensor Decomposition and Functional Data Registration](https://mlsanity.com/paper/2608.02576/) (2026)

## Abstract (arXiv)

In this paper, we provide novel algorithms with identifiability guarantees for simplex-structured matrix factorization (SSMF), a generalization of nonnegative matrix factorization. Current state-of-the-art algorithms that provide identifiability results for SSMF rely on the sufficiently scattered condition (SSC) which requires the data points to be well spread within the convex hull of the basis vectors. The conditions under which our proposed algorithms recover the unique decomposition is in most cases much weaker than the SSC. We only require to have d points on each facet of the convex hull of the basis vectors whose dimension is d-1. The key idea is based on extracting facets containing the largest number of points. We illustrate the effectiveness of our approach on synthetic data sets and hyperspectral images, showing that it outperforms state-of-the-art SSMF algorithms as it is able to handle higher noise levels, rank deficient matrices, outliers, and input data that highly violates the SSC.