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

2025-12-25

AI Summary

This paper addresses the identifiability problem of five nonnegative matrix factorization (NMF) models—LBA, LCA, EMA, PLSA, and standard NMF—under a unified probabilistic modeling, convex geometric analysis, and constrained optimization framework. We rigorously prove, for the first time, that the identifiability of solutions to all five models is *exactly equivalent* to that of standard NMF. This equivalence bridges long-standing theoretical fragmentation across cognitive modeling, machine learning, and statistical inference. Our framework provides the first general, cross-model identifiability criterion. Experiments on real-world time-budget data empirically validate the theoretical predictions and clarify boundary conditions relative to prototype analysis and related methods. The core contribution lies in unifying disparate identifiability theories—previously scattered across disciplines—under the foundational NMF paradigm, thereby establishing a rigorous, model-agnostic basis for principled model selection and interpretability assessment.

Technology Category

• Natural Language Processing: Interpretability, Analysis, and Evaluation of NLP Models
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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.

Problem

• Compares five nonnegative matrix factorization models for identifiability
• Proves equivalence of uniqueness conditions across LBA, EMA, LCA, PLSA, and NMF
• Reviews algorithms and illustrates with a social science dataset

Innovation

• Compares five matrix factorization models for identifiability
• Proves equivalence of uniqueness conditions across these models
• Reviews algorithms and illustrates with social science data