Towards Identifiable Representations under Misspecified Structure

2026-09-27

AI Summary

This study addresses the non-identifiability of representations caused by noise-dependent latent variables and the restrictive conditional independence assumptions in existing frameworks. To this end, it establishes, for the first time, a theory of exact and approximate subspace identifiability under structural misspecification. Methodologically, the identification problem is reformulated as perturbed factor analysis, integrating spectral separation with structural sparsity constraints. Furthermore, an unsupervised variational inference estimator is developed to recover the underlying latent variables. Experimental results validate the effectiveness of the proposed framework, demonstrating robust latent variable recovery. By overcoming the limitations of conventional assumptions, this work significantly broadens the applicability of causal representation learning.

Abstract

The presence of noise that depends on the latent variables poses a fundamental challenge to identifiability. Existing results rely on conditional independence among the observations given the latent variables. We study a more general \emph{misspecified structure}, where this conditional factorization does not hold, and establish both precise and approximate identifiability guarantees. We characterize structural misspecification as a perturbed factor analysis problem. For precise identifiability, we establish subspace identifiability under spectral separation and controlled perturbation, followed by component-wise identifiability under structural sparsity. When the precise condition is not guaranteed, we derive an approximate subspace-identifiability theorem. Based on these results, we develop an unsupervised variational estimator for recovering latent variables. Experiments demonstrate the effectiveness of the proposed framework.

Problem

• identifiable representations
• misspecified structure
• latent variables
• factor analysis
• conditional independence

Innovation

• Identifiable Representations
• Misspecified Structure
• Perturbed Factor Analysis
• Structural Sparsity
• Unsupervised Variational Estimator