A review of NMF, PLSA, LBA, EMA, and LCA with a focus on the identifiability issue

TL;DR

This paper reviews five matrix factorization models, highlighting their similarities and the crucial issue of identifiability, and demonstrates the conditions under which their solutions are unique.

Contribution

It establishes the equivalence of solution uniqueness among LBA, EMA, LCA, PLSA, and NMF models, and reviews algorithms and applications of these models.

Findings

Solution of LBA, EMA, LCA, PLSA is unique iff NMF solution is unique.

Models are similar or equivalent despite different presentations.

Illustrated with social science time budget data.

Abstract

Across fields such as machine learning, social science, geography, considerable attention has been given to models that factorize a nonnegative matrix into the product of two or three matrices, subject to nonnegative or row-sum-to-1 constraints. Although these models are to a large extend similar or even equivalent, they are presented under different names, and their similarity is not well known. This paper highlights similarities among five popular models, latent budget analysis (LBA), latent class analysis (LCA), end-member analysis (EMA), probabilistic latent semantic analysis (PLSA), and nonnegative matrix factorization (NMF). We focus on an essential issue-identifiability-of these models and prove that the solution of LBA, EMA, LCA, PLSA is unique if and only if the solution of NMF is unique. We also provide a brief review for algorithms of these models. We illustrate the models with a time budget dataset from social science, and end the paper with a discussion of closely related models such as archetypal analysis.