# Linear Causal Representation Learning from Unknown Multi-node Interventions

arXiv: https://arxiv.org/abs/2406.05937 · HTML: https://mlsanity.com/paper/2406.05937/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 39 references that are in the map.
- First posted on arXiv 9 Jun 2024.

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

- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Self-Supervised Learning with Data Augmentations Provably Isolates Content from Style](https://mlsanity.com/paper/2106.04619/) (2021)
- [Interventional Causal Representation Learning](https://mlsanity.com/paper/2209.11924/) (2022)
- [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)
- [Multi-View Causal Representation Learning with Partial Observability](https://mlsanity.com/paper/2311.04056/) (2023)
- [Permutation-Based Causal Structure Learning with Unknown Intervention Targets](https://mlsanity.com/paper/1910.09007/) (2019)
- [Causal Representation Learning from Multiple Distributions: A General Setting](https://mlsanity.com/paper/2402.05052/) (2024)
- [Causal Representation Learning for Instantaneous and Temporal Effects in Interactive Systems](https://mlsanity.com/paper/2206.06169/) (2022)
- [Score-based Causal Representation Learning with Interventions](https://mlsanity.com/paper/2301.08230/) (2023)
- [Nonparametric Partial Disentanglement via Mechanism Sparsity: Sparse Actions, Interventions and Sparse Temporal Dependencies](https://mlsanity.com/paper/2401.04890/) (2024)
- [BISCUIT: Causal Representation Learning from Binary Interactions](https://mlsanity.com/paper/2306.09643/) (2023)
- [Learning nonparametric latent causal graphs with unknown interventions](https://mlsanity.com/paper/2306.02899/) (2023)
- [General Identifiability and Achievability for Causal Representation Learning](https://mlsanity.com/paper/2310.15450/) (2023)
- [Unpaired Multi-Domain Causal Representation Learning](https://mlsanity.com/paper/2302.00993/) (2023)
- [Causal Representation Learning Made Identifiable by Grouping of Observational Variables](https://mlsanity.com/paper/2310.15709/) (2023)
- [Score-based Causal Representation Learning: Linear and General Transformations](https://mlsanity.com/paper/2402.00849/) (2024)
- [Nonparametric Score Estimators](https://mlsanity.com/paper/2005.10099/) (2020)
- [Multi-Domain Causal Representation Learning via Weak Distributional Invariances](https://mlsanity.com/paper/2310.02854/) (2023)
- [Causal machine learning for single-cell genomics](https://mlsanity.com/paper/2310.14935/) (2023)
- [Identifiable Latent Polynomial Causal Models Through the Lens of Change](https://mlsanity.com/paper/2310.15580/) (2023)
- [From Identifiable Causal Representations to Controllable Counterfactual Generation: A Survey on Causal Generative Modeling](https://mlsanity.com/paper/2310.11011/) (2023)
- [Identifying Linearly-Mixed Causal Representations from Multi-Node Interventions](https://mlsanity.com/paper/2311.02695/) (2023)
- [Learning Causal Representations from General Environments: Identifiability and Intrinsic Ambiguity](https://mlsanity.com/paper/2311.12267/) (2023)

## Built on by (most cited first, 14 of 14)

- [Unifying Causal Representation Learning with the Invariance Principle](https://mlsanity.com/paper/2409.02772/) (2024)
- [Debiasing Reward Models by Representation Learning with Guarantees](https://mlsanity.com/paper/2510.23751/) (2025)
- [Causal Representation Learning from General Environments under Nonparametric Mixing](https://mlsanity.com/paper/2604.23800/) (2026)
- [Disentangling Dynamical Systems: Causal Representation Learning Meets Local Sparse Attention](https://mlsanity.com/paper/2603.14483/) (2026)
- [Discovering What You Can Control: Interventional Boundary Discovery for Reinforcement Learning](https://mlsanity.com/paper/2603.18257/) (2026)
- [A General Representation-Based Approach to Multi-Source Domain Adaptation](https://mlsanity.com/paper/2604.23790/) (2026)
- [Concept Modulation Models: A Unified Framework for Identifiability and Extrapolation](https://mlsanity.com/paper/2606.18509/) (2026)
- [Catastrophic Compositional Generation: Why Vanilla Diffusion Models Fail to Extrapolate](https://mlsanity.com/paper/2606.23920/) (2026)
- [Incorporating Interventional Independence Improves Robustness against Interventional Distribution Shift](https://mlsanity.com/paper/2507.05412/) (2025)
- [Learning Robust Intervention Representations with Delta Embeddings](https://mlsanity.com/paper/2508.04492/) (2025)
- [Linear Causal Representation Learning by Topological Ordering, Pruning, and Disentanglement](https://mlsanity.com/paper/2509.22553/) (2025)
- [ROPES: Robotic Pose Estimation via Score-Based Causal Representation Learning](https://mlsanity.com/paper/2510.20884/) (2025)
- [CURVE: Learning Causality-Inspired Invariant Representations for Robust Scene Understanding via Uncertainty-Guided Regularization](https://mlsanity.com/paper/2601.20355/) (2026)
- [Discrete Causal Representations from Heterogeneous Domains: A Bayesian Approach with Social Survey Applications](https://mlsanity.com/paper/2606.06288/) (2026)

## Abstract (arXiv)

Despite the multifaceted recent advances in interventional causal representation learning (CRL), they primarily focus on the stylized assumption of single-node interventions. This assumption is not valid in a wide range of applications, and generally, the subset of nodes intervened in an interventional environment is fully unknown. This paper focuses on interventional CRL under unknown multi-node (UMN) interventional environments and establishes the first identifiability results for general latent causal models (parametric or nonparametric) under stochastic interventions (soft or hard) and linear transformation from the latent to observed space. Specifically, it is established that given sufficiently diverse interventional environments, (i) identifiability up to ancestors is possible using only soft interventions, and (ii) perfect identifiability is possible using hard interventions. Remarkably, these guarantees match the best-known results for more restrictive single-node interventions. Furthermore, CRL algorithms are also provided that achieve the identifiability guarantees. A central step in designing these algorithms is establishing the relationships between UMN interventional CRL and score functions associated with the statistical models of different interventional environments. Establishing these relationships also serves as constructive proof of the identifiability guarantees.