# Beyond ICA: Identifiability by Symmetry Breaking

arXiv: https://arxiv.org/abs/2607.23182 · HTML: https://mlsanity.com/paper/2607.23182/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 46 references that are in the map.
- First posted on arXiv 25 Jul 2026.

## Builds on (most cited first, 21 of 21)

- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [Unsupervised Feature Extraction by Time-Contrastive Learning and Nonlinear ICA](https://mlsanity.com/paper/1605.06336/) (2016)
- [On Linear Identifiability of Learned Representations](https://mlsanity.com/paper/2007.00810/) (2020)
- [Independent mechanism analysis, a new concept?](https://mlsanity.com/paper/2106.05200/) (2021)
- [The Incomplete Rosetta Stone Problem: Identifiability Results for Multi-View Nonlinear ICA](https://mlsanity.com/paper/1905.06642/) (2019)
- [Identifiability of deep generative models without auxiliary information](https://mlsanity.com/paper/2206.10044/) (2022)
- [Additive Decoders for Latent Variables Identification and Cartesian-Product Extrapolation](https://mlsanity.com/paper/2307.02598/) (2023)
- [Deep Neural Networks Learn Non-Smooth Functions Effectively](https://mlsanity.com/paper/1802.04474/) (2018)
- [Function Classes for Identifiable Nonlinear Independent Component Analysis](https://mlsanity.com/paper/2208.06406/) (2022)
- [Provably Learning Object-Centric Representations](https://mlsanity.com/paper/2305.14229/) (2023)
- [Generalizing Nonlinear ICA Beyond Structural Sparsity](https://mlsanity.com/paper/2311.00866/) (2023)
- [Indeterminacy in Generative Models: Characterization and Strong Identifiability](https://mlsanity.com/paper/2206.00801/) (2022)
- [Properties from Mechanisms: An Equivariance Perspective on Identifiable Representation Learning](https://mlsanity.com/paper/2110.15796/) (2021)
- [A State Space Approach for Piecewise-Linear Recurrent Neural Networks for Reconstructing Nonlinear Dynamics from Neural Measurements](https://mlsanity.com/paper/1612.07846/) (2016)
- [Interaction Asymmetry: A General Principle for Learning Composable Abstractions](https://mlsanity.com/paper/2411.07784/) (2024)
- [Toward the Identifiability of Comparative Deep Generative Models](https://mlsanity.com/paper/2401.15903/) (2024)
- [Mechanistic Independence: A Principle for Identifiable Disentangled Representations](https://mlsanity.com/paper/2509.22196/) (2025)
- [Identifiability of Potentially Degenerate Gaussian Mixture Models With Piecewise Affine Mixing](https://mlsanity.com/paper/2604.13218/) (2026)

## Abstract (arXiv)

We prove the identifiability of deep generative models (DGMs) with piecewise-affine (PWA) decoders and Gaussian mixture model (GMM) priors, in a purely unsupervised setting. We introduce three algebraic contrast principles for symmetry breaking: domain contrast, which trivializes the mixture symmetry group; mechanism contrast, which ensures every decoder branch is witnessed by a unique boundary; and interaction contrast, which forbids parameter conspiracies between latent components and decoder branches. Together they exploit the interplay between the discrete combinatorics of the PWA map and the continuous symmetry structure of the latent GMM. Continuity is replaced by algebraic symmetry conditions; injectivity is decoupled from structural identification and required only for pointwise inversion. Our results form a hierarchy: from law identifiability (LID; latent distribution up to a global affine map) through map identifiability (MID; decoder up to the same map) to posterior and pointwise identifiability. The ICA-form ambiguity emerges under conditions on diagonal component covariances. Assumptions are only on the data-generating process, not on learning methods, except for the interaction contrast. To our knowledge this is the first to make algebraic symmetry-breaking the engine of nonlinear identifiability, the first to admit discontinuous decoders, and the first to handle fully non-injective decoders, where every observation admits multiple latent codes.