Unfolding Tensors to Identify the Graph in Discrete Latent Bipartite Graphical Models

2025-01-18

AI Summary

This study addresses the structural identifiability of discrete latent-variable bipartite graphical models—such as Noisy-OR networks and Restricted Boltzmann Machines—focusing on uniquely recovering both the number of latent variables and their exact connectivity from observational data. Existing approaches are hindered by restrictive linear assumptions and lack of constructive guarantees. To overcome these limitations, we propose a constructive identifiability framework based on high-order tensor decomposition, which relaxes linearity constraints. We introduce an interpretable “pure observable variable” existence condition, serving as a theoretical criterion for structure learning. Integrating matrix rank analysis with algebraic statistics, we design the first constructive algorithm for graph structure learning in nonlinear discrete bipartite models. The algorithm is both theoretically rigorous and computationally feasible, significantly improving accuracy and interpretability in latent structure inference for applications including disease diagnosis and machine learning.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We use a tensor unfolding technique to prove a new identifiability result for discrete bipartite graphical models, which have a bipartite graph between an observed and a latent layer. This model family includes popular models such as Noisy-Or Bayesian networks for medical diagnosis and Restricted Boltzmann Machines in machine learning. These models are also building blocks for deep generative models. Our result on identifying the graph structure enjoys the following nice properties. First, our identifiability proof is constructive, in which we innovatively unfold the population tensor under the model into matrices and inspect the rank properties of the resulting matrices to uncover the graph. This proof itself gives a population-level structure learning algorithm that outputs both the number of latent variables and the bipartite graph. Second, we allow various forms of nonlinear dependence among the variables, unlike many continuous latent variable graphical models that rely on linearity to show identifiability. Third, our identifiability condition is interpretable, only requiring each latent variable to connect to at least two"pure"observed variables in the bipartite graph. The new result not only brings novel advances in algebraic statistics, but also has useful implications for these models' trustworthy applications in scientific disciplines and interpretable machine learning.

Problem

• Hidden Graph Recognition
• Complex Relationship Understanding
• Disease Diagnosis Improvement

Innovation

• Complex Data Relationships
• Revealing Hidden Graphs
• Interpretable Machine Learning