# Learning latent causal graphs via mixture oracles

arXiv: https://arxiv.org/abs/2106.15563 · HTML: https://mlsanity.com/paper/2106.15563/

## Key facts

- Cited by 55 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 99 references that are in the map.
- First posted on arXiv 29 Jun 2021.

## Builds on (most cited first, 28 of 28)

- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [Causal Inference in the Presence of Latent Variables and Selection Bias](https://mlsanity.com/paper/1302.4983/) (2013)
- [CausalGAN: Learning Causal Implicit Generative Models with Adversarial Training](https://mlsanity.com/paper/1709.02023/) (2017)
- [Discovering Causal Signals in Images](https://mlsanity.com/paper/1605.08179/) (2016)
- [Causal Discovery in Physical Systems from Videos](https://mlsanity.com/paper/2007.00631/) (2020)
- [Visual Causal Feature Learning](https://mlsanity.com/paper/1412.2309/) (2014)
- [Pixels to Graphs by Associative Embedding](https://mlsanity.com/paper/1706.07365/) (2017)
- [Causal Discovery with General Non-Linear Relationships Using Non-Linear ICA](https://mlsanity.com/paper/1904.09096/) (2019)
- [Nested Markov Properties for Acyclic Directed Mixed Graphs](https://mlsanity.com/paper/1701.06686/) (2017)
- [A polynomial-time algorithm for learning nonparametric causal graphs](https://mlsanity.com/paper/2006.11970/) (2020)
- [Margins of discrete Bayesian networks](https://mlsanity.com/paper/1501.02103/) (2015)
- [On Multi-Cause Causal Inference with Unobserved Confounding: Counterexamples, Impossibility, and Alternatives](https://mlsanity.com/paper/1902.10286/) (2019)
- [Robust causal structure learning with some hidden variables](https://mlsanity.com/paper/1708.01151/) (2017)
- [Identifiability of Nonparametric Mixture Models and Bayes Optimal Clustering](https://mlsanity.com/paper/1802.04397/) (2018)
- [Measurement Dependence Inducing Latent Causal Models](https://mlsanity.com/paper/1910.08778/) (2019)
- [Markovian acyclic directed mixed graphs for discrete data](https://mlsanity.com/paper/1301.6624/) (2013)
- [A Uniformly Consistent Estimator of Causal Effects under the k-Triangle-Faithfulness Assumption](https://mlsanity.com/paper/1502.00829/) (2015)
- [Learning Restricted Boltzmann Machines via Influence Maximization](https://mlsanity.com/paper/1805.10262/) (2018)
- [Smooth, identifiable supermodels of discrete DAG models with latent variables](https://mlsanity.com/paper/1511.06813/) (2015)
- [Estimating the Number of Components in Finite Mixture Models via the Group-Sort-Fuse Procedure](https://mlsanity.com/paper/2005.11641/) (2020)
- [Consistent Estimation of Identifiable Nonparametric Mixture Models from Grouped Observations](https://mlsanity.com/paper/2006.07459/) (2020)
- [Na"ive regression requires weaker assumptions than factor models to adjust for multiple cause confounding](https://mlsanity.com/paper/2007.12702/) (2020)
- [Hadamard Extensions and the Identification of Mixtures of Product Distributions](https://mlsanity.com/paper/2101.11688/) (2021)
- [Finite state space non parametric Hidden Markov Models are in general identifiable](https://mlsanity.com/paper/1306.4657/) (2013)
- [Smoothed Analysis of Discrete Tensor Decomposition and Assemblies of Neurons](https://mlsanity.com/paper/1810.11896/) (2018)
- [Efficient Tensor Decomposition](https://mlsanity.com/paper/2007.15589/) (2020)

## Built on by (most cited first, 30 of 55)

- [Weakly supervised causal representation learning](https://mlsanity.com/paper/2203.16437/) (2022)
- [Nonparametric Identifiability of Causal Representations from Unknown Interventions](https://mlsanity.com/paper/2306.00542/) (2023)
- [Learning Linear Causal Representations from Interventions under General Nonlinear Mixing](https://mlsanity.com/paper/2306.02235/) (2023)
- [Linear Causal Disentanglement via Interventions](https://mlsanity.com/paper/2211.16467/) (2022)
- [Identifiability Guarantees for Causal Disentanglement from Soft Interventions](https://mlsanity.com/paper/2307.06250/) (2023)
- [On the Origins of Linear Representations in Large Language Models](https://mlsanity.com/paper/2403.03867/) (2024)
- [Multi-View Causal Representation Learning with Partial Observability](https://mlsanity.com/paper/2311.04056/) (2023)
- [Identifiability of deep generative models without auxiliary information](https://mlsanity.com/paper/2206.10044/) (2022)
- [Causal Representation Learning from Multiple Distributions: A General Setting](https://mlsanity.com/paper/2402.05052/) (2024)
- [Latent Hierarchical Causal Structure Discovery with Rank Constraints](https://mlsanity.com/paper/2210.01798/) (2022)
- [Causal Structure Learning: a Combinatorial Perspective](https://mlsanity.com/paper/2206.01152/) (2022)
- [A Sparsity Principle for Partially Observable Causal Representation Learning](https://mlsanity.com/paper/2403.08335/) (2024)
- [Identifying Weight-Variant Latent Causal Models](https://mlsanity.com/paper/2208.14153/) (2022)
- [Learning nonparametric latent causal graphs with unknown interventions](https://mlsanity.com/paper/2306.02899/) (2023)
- [A Versatile Causal Discovery Framework to Allow Causally-Related Hidden Variables](https://mlsanity.com/paper/2312.11001/) (2023)
- [Causal Representation Learning Made Identifiable by Grouping of Observational Variables](https://mlsanity.com/paper/2310.15709/) (2023)
- [Learning Interpretable Concepts: Unifying Causal Representation Learning and Foundation Models](https://mlsanity.com/paper/2402.09236/) (2024)
- [Identification of Nonlinear Latent Hierarchical Models](https://mlsanity.com/paper/2306.07916/) (2023)
- [Identifying Representations for Intervention Extrapolation](https://mlsanity.com/paper/2310.04295/) (2023)
- [Learning Discrete Concepts in Latent Hierarchical Models](https://mlsanity.com/paper/2406.00519/) (2024)
- [I Predict Therefore I Am: Is Next Token Prediction Enough to Learn Human-Interpretable Concepts from Data?](https://mlsanity.com/paper/2503.08980/) (2025)
- [Towards Interpretable Deep Generative Models via Causal Representation Learning](https://mlsanity.com/paper/2504.11609/) (2025)
- [Smoke and Mirrors in Causal Downstream Tasks](https://mlsanity.com/paper/2405.17151/) (2024)
- [Identifiable Causal Representation Learning: Unsupervised, Multi-View, and Multi-Environment](https://mlsanity.com/paper/2406.13371/) (2024)
- [Causal Inference Despite Limited Global Confounding via Mixture Models](https://mlsanity.com/paper/2112.11602/) (2021)
- [An Interventional Perspective on Identifiability in Gaussian LTI Systems with Independent Component Analysis](https://mlsanity.com/paper/2311.18048/) (2023)
- [Tier Balancing: Towards Dynamic Fairness over Underlying Causal Factors](https://mlsanity.com/paper/2301.08987/) (2023)
- [A Skewness-Based Criterion for Addressing Heteroscedastic Noise in Causal Discovery](https://mlsanity.com/paper/2410.06407/) (2024)
- [Identifiability Guarantees for Causal Disentanglement from Purely Observational Data](https://mlsanity.com/paper/2410.23620/) (2024)
- [Latent Variable Causal Discovery under Selection Bias](https://mlsanity.com/paper/2512.11219/) (2025)

## Abstract (arXiv)

We study the problem of reconstructing a causal graphical model from data in the presence of latent variables. The main problem of interest is recovering the causal structure over the latent variables while allowing for general, potentially nonlinear dependence between the variables. In many practical problems, the dependence between raw observations (e.g. pixels in an image) is much less relevant than the dependence between certain high-level, latent features (e.g. concepts or objects), and this is the setting of interest. We provide conditions under which both the latent representations and the underlying latent causal model are identifiable by a reduction to a mixture oracle. These results highlight an intriguing connection between the well-studied problem of learning the order of a mixture model and the problem of learning the bipartite structure between observables and unobservables. The proof is constructive, and leads to several algorithms for explicitly reconstructing the full graphical model. We discuss efficient algorithms and provide experiments illustrating the algorithms in practice.