# Latent Hierarchical Causal Structure Discovery with Rank Constraints

arXiv: https://arxiv.org/abs/2210.01798 · HTML: https://mlsanity.com/paper/2210.01798/

## Key facts

- Cited by 51 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 48 references that are in the map.
- First posted on arXiv 1 Oct 2022.

## Builds on (most cited first, 11 of 11)

- [Causal Discovery from Heterogeneous/Nonstationary Data with Independent Changes](https://mlsanity.com/paper/1903.01672/) (2019)
- [Learning latent causal graphs via mixture oracles](https://mlsanity.com/paper/2106.15563/) (2021)
- [Learning Linear Non-Gaussian Causal Models in the Presence of Latent Variables](https://mlsanity.com/paper/1908.03932/) (2019)
- [ParceLiNGAM: A causal ordering method robust against latent confounders](https://mlsanity.com/paper/1303.7410/) (2013)
- [Recursive Causal Structure Learning in the Presence of Latent Variables and Selection Bias](https://mlsanity.com/paper/2110.12036/) (2021)
- [Causal Structure Discovery between Clusters of Nodes Induced by Latent Factors](https://mlsanity.com/paper/2207.01237/) (2022)
- [Robust causal structure learning with some hidden variables](https://mlsanity.com/paper/1708.01151/) (2017)
- [Causal Discovery with Multi-Domain LiNGAM for Latent Factors](https://mlsanity.com/paper/2009.09176/) (2020)
- [Calculation of Entailed Rank Constraints in Partially Non-Linear and Cyclic Models](https://mlsanity.com/paper/1309.7004/) (2013)
- [Marginal likelihood and model selection for Gaussian latent tree and forest models](https://mlsanity.com/paper/1412.8285/) (2014)
- [Causal Clustering for 1-Factor Measurement Models on Data with Various Types](https://mlsanity.com/paper/2009.08606/) (2020)

## Built on by (most cited first, 30 of 51)

- [Causal Representation Learning from Multiple Distributions: A General Setting](https://mlsanity.com/paper/2402.05052/) (2024)
- [Understanding Masked Autoencoders via Hierarchical Latent Variable Models](https://mlsanity.com/paper/2306.04898/) (2023)
- [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)
- [Unpaired Multi-Domain Causal Representation Learning](https://mlsanity.com/paper/2302.00993/) (2023)
- [Identification of Nonlinear Latent Hierarchical Models](https://mlsanity.com/paper/2306.07916/) (2023)
- [Identifiable Latent Polynomial Causal Models Through the Lens of Change](https://mlsanity.com/paper/2310.15580/) (2023)
- [CaRiNG: Learning Temporal Causal Representation under Non-Invertible Generation Process](https://mlsanity.com/paper/2401.14535/) (2024)
- [Learning Discrete Concepts in Latent Hierarchical Models](https://mlsanity.com/paper/2406.00519/) (2024)
- [On the Identification of Temporally Causal Representation with Instantaneous Dependence](https://mlsanity.com/paper/2405.15325/) (2024)
- [Causal Temporal Representation Learning with Nonstationary Sparse Transition](https://mlsanity.com/paper/2409.03142/) (2024)
- [Towards Interpretable Deep Generative Models via Causal Representation Learning](https://mlsanity.com/paper/2504.11609/) (2025)
- [Emerging Synergies in Causality and Deep Generative Models: A Survey](https://mlsanity.com/paper/2301.12351/) (2023)
- [Identification of Causal Structure with Latent Variables Based on Higher Order Cumulants](https://mlsanity.com/paper/2312.11934/) (2023)
- [On the Parameter Identifiability of Partially Observed Linear Causal Models](https://mlsanity.com/paper/2407.16975/) (2024)
- [A Skewness-Based Criterion for Addressing Heteroscedastic Noise in Causal Discovery](https://mlsanity.com/paper/2410.06407/) (2024)
- [Latent Variable Causal Discovery under Selection Bias](https://mlsanity.com/paper/2512.11219/) (2025)
- [Debiasing Reward Models by Representation Learning with Guarantees](https://mlsanity.com/paper/2510.23751/) (2025)
- [Towards Identifiable Latent Additive Noise Models](https://mlsanity.com/paper/2403.15711/) (2024)
- [Identifiability Analysis of Linear ODE Systems with Hidden Confounders](https://mlsanity.com/paper/2410.21917/) (2024)
- [Permutation-Based Rank Test in the Presence of Discretization and Application in Causal Discovery with Mixed Data](https://mlsanity.com/paper/2501.18990/) (2025)
- [Causal Representation Learning from General Environments under Nonparametric Mixing](https://mlsanity.com/paper/2604.23800/) (2026)
- [Neuro-Causal Factor Analysis](https://mlsanity.com/paper/2305.19802/) (2023)
- [Towards Characterizing Domain Counterfactuals For Invertible Latent Causal Models](https://mlsanity.com/paper/2306.11281/) (2023)
- [Causal Discovery in Mixtures of Populations](https://mlsanity.com/paper/2311.07454/) (2023)
- [Long-Term Fairness Inquiries and Pursuits in Machine Learning: A Survey of Notions, Methods, and Challenges](https://mlsanity.com/paper/2406.06736/) (2024)
- [CausalVerse: Benchmarking Causal Representation Learning with Configurable High-Fidelity Simulations](https://mlsanity.com/paper/2510.14049/) (2025)
- [Distributional Equivalence in Linear Non-Gaussian Latent-Variable Cyclic Causal Models: Characterization and Learning](https://mlsanity.com/paper/2603.04780/) (2026)
- [s-ID: Causal Effect Identification in a Sub-Population](https://mlsanity.com/paper/2309.02281/) (2023)
- [Learning Discrete Latent Variable Structures with Tensor Rank Conditions](https://mlsanity.com/paper/2406.07020/) (2024)

## Abstract (arXiv)

Most causal discovery procedures assume that there are no latent confounders in the system, which is often violated in real-world problems. In this paper, we consider a challenging scenario for causal structure identification, where some variables are latent and they form a hierarchical graph structure to generate the measured variables; the children of latent variables may still be latent and only leaf nodes are measured, and moreover, there can be multiple paths between every pair of variables (i.e., it is beyond tree structure). We propose an estimation procedure that can efficiently locate latent variables, determine their cardinalities, and identify the latent hierarchical structure, by leveraging rank deficiency constraints over the measured variables. We show that the proposed algorithm can find the correct Markov equivalence class of the whole graph asymptotically under proper restrictions on the graph structure.