Partial Identifiability for Nonnegative Matrix Factorization
| Autor: | Nicolas Gillis, Róbert Rajkó |
|---|---|
| Rok vydání: | 2023 |
| Předmět: |
Signal Processing (eess.SP)
FOS: Computer and information sciences Computer Science - Machine Learning Statistics - Machine Learning FOS: Mathematics FOS: Electrical engineering electronic engineering information engineering Machine Learning (stat.ML) Numerical Analysis (math.NA) Mathematics - Numerical Analysis Electrical Engineering and Systems Science - Signal Processing Analysis Machine Learning (cs.LG) |
| Zdroj: | SIAM Journal on Matrix Analysis and Applications. 44:27-52 |
| ISSN: | 1095-7162 0895-4798 |
| DOI: | 10.1137/22m1507553 |
| Popis: | 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. 27 pages, 8 figures, 7 examples. This third version makes minor modifications. Paper accepted in SIAM J. on Matrix Analysis and Applications |
| Databáze: | OpenAIRE |
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