On Identifiability of Nonnegative Matrix Factorization

TL;DR

This paper introduces a new criterion for identifying low-rank factors in nonnegative matrix factorization, requiring only mild conditions like sufficient scattering of one factor's rows, without structural assumptions on the other.

Contribution

It proposes a novel identification criterion that guarantees NMF factor recovery under the mildest known conditions, expanding the theoretical understanding of NMF identifiability.

Findings

Guarantees recovery of latent factors under mild conditions

Requires only sufficient scattering of one factor's rows

No structural assumptions needed on the other factor

Abstract

In this letter, we propose a new identification criterion that guarantees the recovery of the low-rank latent factors in the nonnegative matrix factorization (NMF) model, under mild conditions. Specifically, using the proposed criterion, it suffices to identify the latent factors if the rows of one factor are \emph{sufficiently scattered} over the nonnegative orthant, while no structural assumption is imposed on the other factor except being full-rank. This is by far the mildest condition under which the latent factors are provably identifiable from the NMF model.