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This research advances theoretical foundations in unsupervised learning by establishing identifiability guarantees under realistic conditions where noise depends on latent variables and standard conditional independence assumptions fail. By framing structural misspecification as a perturbed factor analysis problem, the authors provide both precise subspace recovery methods and approximate estimators for scenarios with spectral perturbations or sparsity constraints. The work offers critical insights for practitioners deploying variational models in noisy environments where traditional identifiability proofs do not apply.
Towards Identifiable Representations under Misspecified Structure
arXiv:2609.33273v1 Announce Type: new 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.
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.
Machine Learning (cs.LG)
arXiv:2609.33273 [cs.LG]
(or arXiv:2609.33273v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.33273 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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