🤖 AI Summary
This work addresses the identifiability of disentangled representations without relying on strong distributional assumptions—such as statistical independence—thereby establishing necessary and sufficient conditions for unique recovery of latent factors under nonlinear, non-invertible mixing.
Method: We introduce “mechanism independence” as a novel paradigm, modeling latent variables via their generative mechanisms rather than latent distributions. We develop a spectrum of identifiability criteria grounded in support-set structure, sparsity, and higher-order conditional independence, and employ graph-theoretic analysis to characterize connected components of latent subspaces.
Contribution/Results: We prove that each mechanism independence condition guarantees uniqueness of the latent subspace. Our framework transcends classical assumptions in causal discovery and representation learning—namely linearity, invertibility, or statistical independence—and provides the first rigorous, assumption-free theoretical foundation for unsupervised disentanglement.
📝 Abstract
Disentangled representations seek to recover latent factors of variation underlying observed data, yet their identifiability is still not fully understood. We introduce a unified framework in which disentanglement is achieved through mechanistic independence, which characterizes latent factors by how they act on observed variables rather than by their latent distribution. This perspective is invariant to changes of the latent density, even when such changes induce statistical dependencies among factors. Within this framework, we propose several related independence criteria -- ranging from support-based and sparsity-based to higher-order conditions -- and show that each yields identifiability of latent subspaces, even under nonlinear, non-invertible mixing. We further establish a hierarchy among these criteria and provide a graph-theoretic characterization of latent subspaces as connected components. Together, these results clarify the conditions under which disentangled representations can be identified without relying on statistical assumptions.