# Bounded Simplex-Structured Matrix Factorization: Algorithms, Identifiability and Applications

arXiv: https://arxiv.org/abs/2209.12638 · HTML: https://mlsanity.com/paper/2209.12638/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 52 references that are in the map.
- First posted on arXiv 26 Sep 2022.

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

- [Variational Autoencoders for Collaborative Filtering](https://mlsanity.com/paper/1802.05814/) (2018)
- [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)
- [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)
- [Revisiting the Performance of iALS on Item Recommendation Benchmarks](https://mlsanity.com/paper/2110.14037/) (2021)
- [An Inertial Block Majorization Minimization Framework for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2010.12133/) (2020)
- [Learning a Latent Simplex in Input-Sparsity Time](https://mlsanity.com/paper/2105.08005/) (2021)
- [Algorithms and Comparisons of Non-negative Matrix Factorization with Volume Regularization for Hyperspectral Unmixing](https://mlsanity.com/paper/1903.04362/) (2019)
- [Exact and Heuristic Algorithms for Semi-Nonnegative Matrix Factorization](https://mlsanity.com/paper/1410.7220/) (2014)
- [Multiplicative Updates for NMF with β-Divergences under Disjoint Equality Constraints](https://mlsanity.com/paper/2010.16223/) (2020)
- [Binary Component Decomposition Part I: The Positive-Semidefinite Case](https://mlsanity.com/paper/1907.13603/) (2019)
- [Accelerating Block Coordinate Descent for Nonnegative Tensor Factorization](https://mlsanity.com/paper/2001.04321/) (2020)

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

- [Checking the Sufficiently Scattered Condition using a Global Non-Convex Optimization Software](https://mlsanity.com/paper/2402.06019/) (2024)
- [Alternating Direction Method of Multipliers for Nonlinear Matrix Decompositions](https://mlsanity.com/paper/2512.17473/) (2025)
- [Subspace-Constrained Quadratic Matrix Factorization: Algorithm and Applications](https://mlsanity.com/paper/2411.04717/) (2024)
- [From latent factors to language: a user study on LLM-generated explanations for an inherently interpretable matrix-based recommender system](https://mlsanity.com/paper/2509.18980/) (2025)
- [Subzero matrix completion for sparse data analysis: large-scale learning of latent low-rank structure](https://mlsanity.com/paper/2608.21607/) (2026)

## Abstract (arXiv)

In this paper, we propose a new low-rank matrix factorization model dubbed bounded simplex-structured matrix factorization (BSSMF). Given an input matrix X and a factorization rank r, BSSMF looks for a matrix W with r columns and a matrix H with r rows such that X ≈ WH where the entries in each column of W are bounded, that is, they belong to given intervals, and the columns of H belong to the probability simplex, that is, H is column stochastic. BSSMF generalizes nonnegative matrix factorization (NMF), and simplex-structured matrix factorization (SSMF). BSSMF is particularly well suited when the entries of the input matrix X belong to a given interval; for example when the rows of X represent images, or X is a rating matrix such as in the Netflix and MovieLens datasets where the entries of X belong to the interval [1,5]. The simplex-structured matrix H not only leads to an easily understandable decomposition providing a soft clustering of the columns of X, but implies that the entries of each column of WH belong to the same intervals as the columns of W. In this paper, we first propose a fast algorithm for BSSMF, even in the presence of missing data in X. Then we provide identifiability conditions for BSSMF, that is, we provide conditions under which BSSMF admits a unique decomposition, up to trivial ambiguities. Finally, we illustrate the effectiveness of BSSMF on two applications: extraction of features in a set of images, and the matrix completion problem for recommender systems.