# Disentanglement of Correlated Factors via Hausdorff Factorized Support

arXiv: https://arxiv.org/abs/2210.07347 · HTML: https://mlsanity.com/paper/2210.07347/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 65 references that are in the map.
- First posted on arXiv 13 Oct 2022.

## Builds on (most cited first, 30 of 40)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [WILDS: A Benchmark of in-the-Wild Distribution Shifts](https://mlsanity.com/paper/2012.07421/) (2020)
- [VICReg: Variance-Invariance-Covariance Regularization for Self-Supervised Learning](https://mlsanity.com/paper/2105.04906/) (2021)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [GIRAFFE: Representing Scenes as Compositional Generative Neural Feature Fields](https://mlsanity.com/paper/2011.12100/) (2020)
- [Recognition in Terra Incognita](https://mlsanity.com/paper/1807.04975/) (2018)
- [Disentangling by Factorising](https://mlsanity.com/paper/1802.05983/) (2018)
- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [Variational Inference of Disentangled Latent Concepts from Unlabeled Observations](https://mlsanity.com/paper/1711.00848/) (2017)
- [Weakly-Supervised Disentanglement Without Compromises](https://mlsanity.com/paper/2002.02886/) (2020)
- [On the Fairness of Disentangled Representations](https://mlsanity.com/paper/1905.13662/) (2019)
- [Recent Advances in Autoencoder-Based Representation Learning](https://mlsanity.com/paper/1812.05069/) (2018)
- [A Variational U-Net for Conditional Appearance and Shape Generation](https://mlsanity.com/paper/1804.04694/) (2018)
- [Towards Nonlinear Disentanglement in Natural Data with Temporal Sparse Coding](https://mlsanity.com/paper/2007.10930/) (2020)
- [Robustly Disentangled Causal Mechanisms: Validating Deep Representations for Interventional Robustness](https://mlsanity.com/paper/1811.00007/) (2018)
- [Learning Deep Disentangled Embeddings with the F-Statistic Loss](https://mlsanity.com/paper/1802.05312/) (2018)
- [Open World Compositional Zero-Shot Learning](https://mlsanity.com/paper/2101.12609/) (2021)
- [Multi-Level Variational Autoencoder: Learning Disentangled Representations from Grouped Observations](https://mlsanity.com/paper/1705.08841/) (2017)
- [On the Transfer of Inductive Bias from Simulation to the Real World: a New Disentanglement Dataset](https://mlsanity.com/paper/1906.03292/) (2019)
- [Revisiting Training Strategies and Generalization Performance in Deep Metric Learning](https://mlsanity.com/paper/2002.08473/) (2020)
- [On Disentangled Representations Learned From Correlated Data](https://mlsanity.com/paper/2006.07886/) (2020)
- [Variational Autoencoders Pursue PCA Directions (by Accident)](https://mlsanity.com/paper/1812.06775/) (2018)
- [Desiderata for Representation Learning: A Causal Perspective](https://mlsanity.com/paper/2109.03795/) (2021)
- [On the Transfer of Disentangled Representations in Realistic Settings](https://mlsanity.com/paper/2010.14407/) (2020)
- [A Sober Look at the Unsupervised Learning of Disentangled Representations and their Evaluation](https://mlsanity.com/paper/2010.14766/) (2020)

## Built on by (most cited first, 30 of 37)

- [Nonparametric Identifiability of Causal Representations from Unknown Interventions](https://mlsanity.com/paper/2306.00542/) (2023)
- [Additive Decoders for Latent Variables Identification and Cartesian-Product Extrapolation](https://mlsanity.com/paper/2307.02598/) (2023)
- [Disentanglement via Latent Quantization](https://mlsanity.com/paper/2305.18378/) (2023)
- [Cross-Entropy Is All You Need To Invert the Data Generating Process](https://mlsanity.com/paper/2410.21869/) (2024)
- [Probabilistic Contrastive Learning Recovers the Correct Aleatoric Uncertainty of Ambiguous Inputs](https://mlsanity.com/paper/2302.02865/) (2023)
- [A Practitioner's Guide to Continual Multimodal Pretraining](https://mlsanity.com/paper/2408.14471/) (2024)
- [Fantastic Gains and Where to Find Them: On the Existence and Prospect of General Knowledge Transfer between Any Pretrained Model](https://mlsanity.com/paper/2310.17653/) (2023)
- [Memory Mosaics](https://mlsanity.com/paper/2405.06394/) (2024)
- [Identifiable Causal Representation Learning: Unsupervised, Multi-View, and Multi-Environment](https://mlsanity.com/paper/2406.13371/) (2024)
- [Graph-based Unsupervised Disentangled Representation Learning via Multimodal Large Language Models](https://mlsanity.com/paper/2407.18999/) (2024)
- [Subspace-Boosted Model Merging](https://mlsanity.com/paper/2506.16506/) (2025)
- [Disentangling representations of retinal images with generative models](https://mlsanity.com/paper/2402.19186/) (2024)
- [Robustness of Nonlinear Representation Learning](https://mlsanity.com/paper/2503.15355/) (2025)
- [Improving Intervention Efficacy via Concept Realignment in Concept Bottleneck Models](https://mlsanity.com/paper/2405.01531/) (2024)
- [Towards the Reusability and Compositionality of Causal Representations](https://mlsanity.com/paper/2403.09830/) (2024)
- [Representational Similarity via Interpretable Visual Concepts](https://mlsanity.com/paper/2503.15699/) (2025)
- [Memory Mosaics at scale](https://mlsanity.com/paper/2507.03285/) (2025)
- [Disentanglement Learning via Topology](https://mlsanity.com/paper/2308.12696/) (2023)
- [Range, not Independence, Drives Modularity in Biologically Inspired Representations](https://mlsanity.com/paper/2410.06232/) (2024)
- [α-TCVAE: On the relationship between Disentanglement and Diversity](https://mlsanity.com/paper/2411.00588/) (2024)
- [On the Identifiability of Quantized Factors](https://mlsanity.com/paper/2306.16334/) (2023)
- [Disentangled Representation Learning with the Gromov-Monge Gap](https://mlsanity.com/paper/2407.07829/) (2024)
- [Correcting Flaws in Common Disentanglement Metrics](https://mlsanity.com/paper/2304.02335/) (2023)
- [Enriching Disentanglement: From Logical Definitions to Quantitative Metrics](https://mlsanity.com/paper/2305.11512/) (2023)
- [Emotion Collider: Dual Hyperbolic Mirror Manifolds for Sentiment Recovery via Anti Emotion Reflection](https://mlsanity.com/paper/2602.16161/) (2026)
- [Revisiting Disentanglement in Downstream Tasks: A Study on Its Necessity for Abstract Visual Reasoning](https://mlsanity.com/paper/2403.00352/) (2024)
- [Disentangled Representation Learning via Modular Compositional Bias](https://mlsanity.com/paper/2510.21402/) (2025)
- [Representation Disentaglement via Regularization by Causal Identification](https://mlsanity.com/paper/2303.00128/) (2023)
- [Modularity Trumps Invariance for Compositional Robustness](https://mlsanity.com/paper/2306.09005/) (2023)
- [Transitivity Recovering Decompositions: Interpretable and Robust Fine-Grained Relationships](https://mlsanity.com/paper/2310.15999/) (2023)

## Abstract (arXiv)

A grand goal in deep learning research is to learn representations capable of generalizing across distribution shifts. Disentanglement is one promising direction aimed at aligning a model's representation with the underlying factors generating the data (e.g. color or background). Existing disentanglement methods, however, rely on an often unrealistic assumption: that factors are statistically independent. In reality, factors (like object color and shape) are correlated. To address this limitation, we consider the use of a relaxed disentanglement criterion -- the Hausdorff Factorized Support (HFS) criterion -- that encourages only pairwise factorized support, rather than a factorial distribution, by minimizing a Hausdorff distance. This allows for arbitrary distributions of the factors over their support, including correlations between them. We show that the use of HFS consistently facilitates disentanglement and recovery of ground-truth factors across a variety of correlation settings and benchmarks, even under severe training correlations and correlation shifts, with in parts over +60% in relative improvement over existing disentanglement methods. In addition, we find that leveraging HFS for representation learning can even facilitate transfer to downstream tasks such as classification under distribution shifts. We hope our original approach and positive empirical results inspire further progress on the open problem of robust generalization. Code available at https://github.com/facebookresearch/disentangling-correlated-factors.