Learning Discrete Latent Variable Structures with Tensor Rank Conditions

TL;DR

This paper introduces a tensor rank-based method for identifying complex discrete latent variable structures and causal relationships from observed data, extending existing boundaries in causal discovery.

Contribution

It proposes a novel tensor rank condition approach for learning discrete latent structures, including non-linear and complex cases, with an accompanying identification algorithm.

Findings

Effective in identifying latent structures in simulated experiments

Extends the boundary of causal discovery with discrete latent variables

Applicable to non-linear and complex latent variable models

Abstract

Unobserved discrete data are ubiquitous in many scientific disciplines, and how to learn the causal structure of these latent variables is crucial for uncovering data patterns. Most studies focus on the linear latent variable model or impose strict constraints on latent structures, which fail to address cases in discrete data involving non-linear relationships or complex latent structures. To achieve this, we explore a tensor rank condition on contingency tables for an observed variable set $\mathbf{X}_p$, showing that the rank is determined by the minimum support of a specific conditional set (not necessary in $\mathbf{X}_p$) that d-separates all variables in $\mathbf{X}_p$. By this, one can locate the latent variable through probing the rank on different observed variables set, and further identify the latent causal structure under some structure assumptions. We present the corresponding identification algorithm and conduct simulated experiments to verify the effectiveness of our method. In general, our results elegantly extend the identification boundary for causal discovery with discrete latent variables and expand the application scope of causal discovery with latent variables.