Partial Identifiability for Nonnegative Matrix Factorization

TL;DR

This paper investigates conditions under which parts of the factors in nonnegative matrix factorization are uniquely identifiable, providing new theorems and geometric insights that extend understanding of partial identifiability.

Contribution

The paper introduces a rigorous theorem for partial identifiability based on simple sparsity and algebraic conditions, and offers geometric interpretations for specific cases, advancing the theory of NMF identifiability.

Findings

A new theorem guarantees partial uniqueness of a single column in NMF.

Geometric interpretation leads to partial identifiability results for r=3.

Sequential application of results can identify more columns in NMF.

Abstract

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.