# Identification of Nonlinear Latent Hierarchical Models

arXiv: https://arxiv.org/abs/2306.07916 · HTML: https://mlsanity.com/paper/2306.07916/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 48 references that are in the map.
- First posted on arXiv 13 Jun 2023.

## Builds on (most cited first, 17 of 17)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Self-Supervised Learning with Data Augmentations Provably Isolates Content from Style](https://mlsanity.com/paper/2106.04619/) (2021)
- [Nonlinear ICA Using Auxiliary Variables and Generalized Contrastive Learning](https://mlsanity.com/paper/1805.08651/) (2018)
- [Disentanglement via Mechanism Sparsity Regularization: A New Principle for Nonlinear ICA](https://mlsanity.com/paper/2107.10098/) (2021)
- [Learning Temporally Causal Latent Processes from General Temporal Data](https://mlsanity.com/paper/2110.05428/) (2021)
- [Learning Linear Causal Representations from Interventions under General Nonlinear Mixing](https://mlsanity.com/paper/2306.02235/) (2023)
- [On the Identifiability of Nonlinear ICA: Sparsity and Beyond](https://mlsanity.com/paper/2206.07751/) (2022)
- [Learning latent causal graphs via mixture oracles](https://mlsanity.com/paper/2106.15563/) (2021)
- [Latent Hierarchical Causal Structure Discovery with Rank Constraints](https://mlsanity.com/paper/2210.01798/) (2022)
- [Understanding Latent Correlation-Based Multiview Learning and Self-Supervision: An Identifiability Perspective](https://mlsanity.com/paper/2106.07115/) (2021)
- [Learning nonparametric latent causal graphs with unknown interventions](https://mlsanity.com/paper/2306.02899/) (2023)
- [Recursive Causal Structure Learning in the Presence of Latent Variables and Selection Bias](https://mlsanity.com/paper/2110.12036/) (2021)
- [Learning Latent Causal Dynamics](https://mlsanity.com/paper/2202.04828/) (2022)
- [Marginal likelihood and model selection for Gaussian latent tree and forest models](https://mlsanity.com/paper/1412.8285/) (2014)
- [Guaranteed Scalable Learning of Latent Tree Models](https://mlsanity.com/paper/1406.4566/) (2014)

## Built on by (most cited first, 27 of 27)

- [Identifiability Guarantees for Causal Disentanglement from Soft Interventions](https://mlsanity.com/paper/2307.06250/) (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)
- [On the Identification of Temporally Causal Representation with Instantaneous Dependence](https://mlsanity.com/paper/2405.15325/) (2024)
- [Counterfactual Generation with Identifiability Guarantees](https://mlsanity.com/paper/2402.15309/) (2024)
- [Towards Interpretable Deep Generative Models via Causal Representation Learning](https://mlsanity.com/paper/2504.11609/) (2025)
- [Identifiability Guarantees for Causal Disentanglement from Purely Observational Data](https://mlsanity.com/paper/2410.23620/) (2024)
- [Learning General Causal Structures with Hidden Dynamic Process for Climate Analysis](https://mlsanity.com/paper/2501.12500/) (2025)
- [From Orthogonality to Dependency: Learning Disentangled Representation for Multi-Modal Time-Series Sensing Signals](https://mlsanity.com/paper/2405.16083/) (2024)
- [Modeling Unseen Environments with Language-guided Composable Causal Components in Reinforcement Learning](https://mlsanity.com/paper/2505.08361/) (2025)
- [The Third Pillar of Causal Analysis? A Measurement Perspective on Causal Representations](https://mlsanity.com/paper/2505.17708/) (2025)
- [The Robustness of Differentiable Causal Discovery in Misspecified Scenarios](https://mlsanity.com/paper/2510.12503/) (2025)
- [Learned Causal Method Prediction](https://mlsanity.com/paper/2311.03989/) (2023)
- [Differentiable Causal Discovery For Latent Hierarchical Causal Models](https://mlsanity.com/paper/2411.19556/) (2024)
- [Nonstationary Time Series Forecasting via Unknown Distribution Adaptation](https://mlsanity.com/paper/2402.12767/) (2024)
- [Towards Generalizable Reinforcement Learning via Causality-Guided Self-Adaptive Representations](https://mlsanity.com/paper/2407.20651/) (2024)
- [Online Time Series Forecasting with Theoretical Guarantees](https://mlsanity.com/paper/2510.18281/) (2025)
- [Time Series Domain Adaptation via Latent Invariant Causal Mechanism](https://mlsanity.com/paper/2502.16637/) (2025)
- [Causal View of Time Series Imputation: Some Identification Results on Missing Mechanism](https://mlsanity.com/paper/2505.07180/) (2025)
- [Learning by Analogy: A Causal Framework for Composition Generalization](https://mlsanity.com/paper/2512.10669/) (2025)
- [Hierarchical Action Learning for Weakly-Supervised Action Segmentation](https://mlsanity.com/paper/2602.24275/) (2026)
- [Identifiability of Potentially Degenerate Gaussian Mixture Models With Piecewise Affine Mixing](https://mlsanity.com/paper/2604.13218/) (2026)
- [Causal Representation Learning from Network Data](https://mlsanity.com/paper/2509.01916/) (2025)
- [Beyond the Black Box: Identifiable Interpretation and Control in Generative Models via Causal Minimality](https://mlsanity.com/paper/2512.10720/) (2025)
- [Diverse Dictionary Learning](https://mlsanity.com/paper/2604.17568/) (2026)
- [Hessian Rank Constraint for Learning Structure of Nonlinear Latent Variable Models](https://mlsanity.com/paper/2609.24241/) (2026)
- [Towards Identifiable Representations under Misspecified Structure](https://mlsanity.com/paper/2609.33273/) (2026)

## Abstract (arXiv)

Identifying latent variables and causal structures from observational data is essential to many real-world applications involving biological data, medical data, and unstructured data such as images and languages. However, this task can be highly challenging, especially when observed variables are generated by causally related latent variables and the relationships are nonlinear. In this work, we investigate the identification problem for nonlinear latent hierarchical causal models in which observed variables are generated by a set of causally related latent variables, and some latent variables may not have observed children. We show that the identifiability of causal structures and latent variables (up to invertible transformations) can be achieved under mild assumptions: on causal structures, we allow for multiple paths between any pair of variables in the graph, which relaxes latent tree assumptions in prior work; on structural functions, we permit general nonlinearity and multi-dimensional continuous variables, alleviating existing work's parametric assumptions. Specifically, we first develop an identification criterion in the form of novel identifiability guarantees for an elementary latent variable model. Leveraging this criterion, we show that both causal structures and latent variables of the hierarchical model can be identified asymptotically by explicitly constructing an estimation procedure. To the best of our knowledge, our work is the first to establish identifiability guarantees for both causal structures and latent variables in nonlinear latent hierarchical models.