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Beyond ICA: Identifiability by Symmetry Breaking
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Proves identifiability (law, map, posterior, pointwise hierarchy) for deep generative models with piecewise-affine decoders and Gaussian mixture latent priors via algebraic symmetry-breaking conditions, without injectivity or continuity.
Problem
Identifiability of nonlinear latent-variable generative models with discontinuous, non-injective piecewise-affine decoders and unobserved-domain (GMM) latent priors, where classical nonlinear ICA relies on observed domain labels, smoothness, and injectivity.
Approach
Replaces differential/continuity arguments with algebraic symmetry collapse: three contrast conditions (mixture symmetry triviality, universal simple boundary, parameter injectivity) turn distributional equality into a global bijection of branch-component pairs, yielding affine identifiability of latent law and decoder.
Key results
- Establishes law identifiability: under parameter injectivity and simple-boundary conditions, the estimator latent GMM equals a global affine transform of the truth latent GMM.
- Establishes map identifiability: under universal simple boundary and mixture symmetry triviality on the truth side, the decoder is recovered up to the same affine map (f = g∘α^{-1}).
- Shows ICA-form ambiguity (α = PΛz + c) under diagonal component covariances (conditional independence).
Surprising findings
- Identifiability holds for discontinuous and fully non-injective PWA decoders; injectivity is needed only for pointwise recovery.
- Under the stated conditions no strict PWA self-map of a GMM can satisfy parameter injectivity, so identifiability fails without it.
Limitations
- Results are purely theoretical with no experiments reported in the provided text.
- Necessity of some conditions (SB, MST, PI) is shown only via examples or conjectured; a full necessity result for PI is left open.
- Gap between chamber-wise and global parameter injectivity is conjectured to be structurally necessary in the non-injective setting.
Experimental setup
Baselines: TCL iVAE Intact-VAE
Builds on: nonlinear ICA iVAE time-contrastive learning independent mechanism analysis
Closest papers by meaning
- Identifiability of deep generative models without auxiliary information (NeurIPS 2022 · similarity 0.90)
- Identifiability and recoverability in self-supervised models (ICLR 2026 · similarity 0.89)
- On the Identifiability of Hybrid Deep Generative Models: Meta-Learning as a Solution (NeurIPS 2024 · similarity 0.87)
- When does LeJEPA learn a World Model? (NeurIPS 2026 · similarity 0.85)
- Robustness of Nonlinear Representation Learning (ICML 2024 · similarity 0.85)
- Exogenous Isomorphism for Counterfactual Identifiability (ICML 2025 · similarity 0.85)
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