# On the Identifiability of Quantized Factors

arXiv: https://arxiv.org/abs/2306.16334 · HTML: https://mlsanity.com/paper/2306.16334/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 50 references that are in the map.
- First posted on arXiv 28 Jun 2023.

## Builds on (most cited first, 29 of 29)

- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [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)
- [Nonlinear ICA Using Auxiliary Variables and Generalized Contrastive Learning](https://mlsanity.com/paper/1805.08651/) (2018)
- [Unsupervised Feature Extraction by Time-Contrastive Learning and Nonlinear ICA](https://mlsanity.com/paper/1605.06336/) (2016)
- [Weakly-Supervised Disentanglement Without Compromises](https://mlsanity.com/paper/2002.02886/) (2020)
- [Disentanglement via Mechanism Sparsity Regularization: A New Principle for Nonlinear ICA](https://mlsanity.com/paper/2107.10098/) (2021)
- [Interventional Causal Representation Learning](https://mlsanity.com/paper/2209.11924/) (2022)
- [Weakly supervised causal representation learning](https://mlsanity.com/paper/2203.16437/) (2022)
- [CITRIS: Causal Identifiability from Temporal Intervened Sequences](https://mlsanity.com/paper/2202.03169/) (2022)
- [Towards Nonlinear Disentanglement in Natural Data with Temporal Sparse Coding](https://mlsanity.com/paper/2007.10930/) (2020)
- [ICE-BeeM: Identifiable Conditional Energy-Based Deep Models Based on Nonlinear ICA](https://mlsanity.com/paper/2002.11537/) (2020)
- [Independent mechanism analysis, a new concept?](https://mlsanity.com/paper/2106.05200/) (2021)
- [On Disentangled Representations Learned From Correlated Data](https://mlsanity.com/paper/2006.07886/) (2020)
- [On the Identifiability of Nonlinear ICA: Sparsity and Beyond](https://mlsanity.com/paper/2206.07751/) (2022)
- [Discovering Causal Signals in Images](https://mlsanity.com/paper/1605.08179/) (2016)
- [Desiderata for Representation Learning: A Causal Perspective](https://mlsanity.com/paper/2109.03795/) (2021)
- [Identifiability of deep generative models without auxiliary information](https://mlsanity.com/paper/2206.10044/) (2022)
- [Weakly Supervised Representation Learning with Sparse Perturbations](https://mlsanity.com/paper/2206.01101/) (2022)
- [Disentangling Identifiable Features from Noisy Data with Structured Nonlinear ICA](https://mlsanity.com/paper/2106.09620/) (2021)
- [On the Transfer of Disentangled Representations in Realistic Settings](https://mlsanity.com/paper/2010.14407/) (2020)
- [Identifiable Deep Generative Models via Sparse Decoding](https://mlsanity.com/paper/2110.10804/) (2021)
- [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)
- [Disentanglement via Latent Quantization](https://mlsanity.com/paper/2305.18378/) (2023)
- [Disentanglement of Correlated Factors via Hausdorff Factorized Support](https://mlsanity.com/paper/2210.07347/) (2022)
- [Towards efficient representation identification in supervised learning](https://mlsanity.com/paper/2204.04606/) (2022)
- [Learning Latent Causal Dynamics](https://mlsanity.com/paper/2202.04828/) (2022)
- [Binary Independent Component Analysis: A Non-stationarity-based Approach](https://mlsanity.com/paper/2111.15431/) (2021)

## Built on by (most cited first, 3 of 3)

- [From superposition to sparse codes: interpretable representations in neural networks](https://mlsanity.com/paper/2503.01824/) (2025)
- [What Drives Compositional Generalization in Visual Generative Models? The Importance of Continuous Training Objectives](https://mlsanity.com/paper/2510.03075/) (2025)
- [Operationalizing Quantized Disentanglement](https://mlsanity.com/paper/2511.20927/) (2025)

## Abstract (arXiv)

Disentanglement aims to recover meaningful latent ground-truth factors from the observed distribution solely, and is formalized through the theory of identifiability. The identifiability of independent latent factors is proven to be impossible in the unsupervised i.i.d. setting under a general nonlinear map from factors to observations. In this work, however, we demonstrate that it is possible to recover quantized latent factors under a generic nonlinear diffeomorphism. We only assume that the latent factors have independent discontinuities in their density, without requiring the factors to be statistically independent. We introduce this novel form of identifiability, termed quantized factor identifiability, and provide a comprehensive proof of the recovery of the quantized factors.