mlsanityData to 7 Oct 2026

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

arXiv 2209.12638 ↗First posted 26 Sep 2022cs.LG cs.IR cs.NAcited by 5

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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.

Key facts. Cited by 5 papers in the map (data through 7 Oct 2026). Builds on 13 of its 52 references that are in the map. First posted on arXiv 26 Sep 2022.

Builds on

13 of 52 references.

References · 13

  1. 2018Variational Autoencoders for Collaborative Filteringcited by 308
  2. 2018Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applicationscited by 55
  3. 2017On Identifiability of Nonnegative Matrix Factorizationcited by 34
  4. 2014Identifiability of the Simplex Volume Minimization Criterion for Blind Hyperspectral Unmixing: The No Pure-Pixel Casecited by 26
  5. 2020Simplex-Structured Matrix Factorization: Sparsity-based Identifiability and Provably Correct Algorithmscited by 21
  6. 2021Revisiting the Performance of iALS on Item Recommendation Benchmarkscited by 21
  7. 2020An Inertial Block Majorization Minimization Framework for Nonsmooth Nonconvex Optimizationcited by 13
  8. 2021Learning a Latent Simplex in Input-Sparsity Timecited by 11
  9. 2019Algorithms and Comparisons of Non-negative Matrix Factorization with Volume Regularization for Hyperspectral Unmixingcited by 10
  10. 2014Exact and Heuristic Algorithms for Semi-Nonnegative Matrix Factorizationcited by 9
  11. 2020Multiplicative Updates for NMF with β-Divergences under Disjoint Equality Constraintscited by 8
  12. 2019Binary Component Decomposition Part I: The Positive-Semidefinite Casecited by 5
1 more
  1. 2020Accelerating Block Coordinate Descent for Nonnegative Tensor Factorizationcited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 2; 2025: 2; 2026: 1.

5 papers.

  1. 2024Checking the Sufficiently Scattered Condition using a Global Non-Convex Optimization Softwarecited by 3
  2. 2025Alternating Direction Method of Multipliers for Nonlinear Matrix Decompositionscited by 3
  3. 2024Subspace-Constrained Quadratic Matrix Factorization: Algorithm and Applicationscited by 1
  4. 2025From latent factors to language: a user study on LLM-generated explanations for an inherently interpretable matrix-based recommender systemcited by 0
  5. 2026Subzero matrix completion for sparse data analysis: large-scale learning of latent low-rank structurecited by 0