# Partial Identifiability for Nonnegative Matrix Factorization

arXiv: https://arxiv.org/abs/2206.08022 · HTML: https://mlsanity.com/paper/2206.08022/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 45 references that are in the map.
- First posted on arXiv 16 Jun 2022.

## Builds on (most cited first, 3 of 3)

- [Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applications](https://mlsanity.com/paper/1803.01257/) (2018)
- [Simplex-Structured Matrix Factorization: Sparsity-based Identifiability and Provably Correct Algorithms](https://mlsanity.com/paper/2007.11446/) (2020)
- [Multiplicative Updates for NMF with β-Divergences under Disjoint Equality Constraints](https://mlsanity.com/paper/2010.16223/) (2020)

## Built on by (most cited first, 3 of 3)

- [Enhancing Uncertainty Estimation and Interpretability via Bayesian Non-negative Decision Layer](https://mlsanity.com/paper/2505.22199/) (2025)
- [Disentangled Generative Graph Representation Learning](https://mlsanity.com/paper/2408.13471/) (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)

## Abstract (arXiv)

Given a nonnegative matrix factorization, R, and a factorization rank, r, Exact nonnegative matrix factorization (Exact NMF) decomposes R as the product of two nonnegative matrices, C and S with r columns, such as R = CS^top. A central research topic in the literature is the conditions under which such a decomposition is unique/identifiable, up to trivial ambiguities. In this paper, we focus on partial identifiability, that is, the uniqueness of a subset of columns of C and S. We start our investigations with the data-based uniqueness (DBU) theorem from the chemometrics literature. The DBU theorem analyzes all feasible solutions of Exact NMF, and relies on sparsity conditions on C and S. We provide a mathematically rigorous theorem of a recently published restricted version of the DBU theorem, relying only on simple sparsity and algebraic conditions: it applies to a particular solution of Exact NMF (as opposed to all feasible solutions) and allows us to guarantee the partial uniqueness of a single column of C or S. Second, based on a geometric interpretation of the restricted DBU theorem, we obtain a new partial identifiability result. This geometric interpretation also leads us to another partial identifiability result in the case r=3. Third, we show how partial identifiability results can be used sequentially to guarantee the identifiability of more columns of C and S. We illustrate these results on several examples, including one from the chemometrics literature.