# Identifying Linearly-Mixed Causal Representations from Multi-Node Interventions

arXiv: https://arxiv.org/abs/2311.02695 · HTML: https://mlsanity.com/paper/2311.02695/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 49 references that are in the map.
- First posted on arXiv 5 Nov 2023.

## Builds on (most cited first, 29 of 29)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (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)
- [Unsupervised Feature Extraction by Time-Contrastive Learning and Nonlinear ICA](https://mlsanity.com/paper/1605.06336/) (2016)
- [Disentanglement via Mechanism Sparsity Regularization: A New Principle for Nonlinear ICA](https://mlsanity.com/paper/2107.10098/) (2021)
- [Interventional Causal Representation Learning](https://mlsanity.com/paper/2209.11924/) (2022)
- [Weakly supervised causal representation learning](https://mlsanity.com/paper/2203.16437/) (2022)
- [CITRIS: Causal Identifiability from Temporal Intervened Sequences](https://mlsanity.com/paper/2202.03169/) (2022)
- [Nonparametric Identifiability of Causal Representations from Unknown Interventions](https://mlsanity.com/paper/2306.00542/) (2023)
- [On Linear Identifiability of Learned Representations](https://mlsanity.com/paper/2007.00810/) (2020)
- [Learning Temporally Causal Latent Processes from General Temporal Data](https://mlsanity.com/paper/2110.05428/) (2021)
- [Independent mechanism analysis, a new concept?](https://mlsanity.com/paper/2106.05200/) (2021)
- [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)
- [Hidden Markov Nonlinear ICA: Unsupervised Learning from Nonstationary Time Series](https://mlsanity.com/paper/2006.12107/) (2020)
- [Learning Causal Graphs with Small Interventions](https://mlsanity.com/paper/1511.00041/) (2015)
- [Temporally Disentangled Representation Learning](https://mlsanity.com/paper/2210.13647/) (2022)
- [Causal Component Analysis](https://mlsanity.com/paper/2305.17225/) (2023)
- [Causal Discovery in Heterogeneous Environments Under the Sparse Mechanism Shift Hypothesis](https://mlsanity.com/paper/2206.02013/) (2022)
- [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)
- [Cost-Optimal Learning of Causal Graphs](https://mlsanity.com/paper/1703.02645/) (2017)
- [Synergies between Disentanglement and Sparsity: Generalization and Identifiability in Multi-Task Learning](https://mlsanity.com/paper/2211.14666/) (2022)
- [BISCUIT: Causal Representation Learning from Binary Interactions](https://mlsanity.com/paper/2306.09643/) (2023)
- [Partial Disentanglement via Mechanism Sparsity](https://mlsanity.com/paper/2207.07732/) (2022)
- [Multi-Domain Causal Representation Learning via Weak Distributional Invariances](https://mlsanity.com/paper/2310.02854/) (2023)
- [Identifying Representations for Intervention Extrapolation](https://mlsanity.com/paper/2310.04295/) (2023)

## Built on by (most cited first, 18 of 18)

- [A Sparsity Principle for Partially Observable Causal Representation Learning](https://mlsanity.com/paper/2403.08335/) (2024)
- [Linear Causal Representation Learning from Unknown Multi-node Interventions](https://mlsanity.com/paper/2406.05937/) (2024)
- [Identifiability Guarantees for Causal Disentanglement from Purely Observational Data](https://mlsanity.com/paper/2410.23620/) (2024)
- [Debiasing Reward Models by Representation Learning with Guarantees](https://mlsanity.com/paper/2510.23751/) (2025)
- [Invariance & Causal Representation Learning: Prospects and Limitations](https://mlsanity.com/paper/2312.03580/) (2023)
- [Sanity Checking Causal Representation Learning on a Simple Real-World System](https://mlsanity.com/paper/2502.20099/) (2025)
- [Causal Representation Learning from General Environments under Nonparametric Mixing](https://mlsanity.com/paper/2604.23800/) (2026)
- [The Causal Information Bottleneck and Optimal Causal Variable Abstractions](https://mlsanity.com/paper/2410.00535/) (2024)
- [Structural Causal Bottleneck Models](https://mlsanity.com/paper/2603.08682/) (2026)
- [Revealing Multimodal Causality with Large Language Models](https://mlsanity.com/paper/2509.17784/) (2025)
- [Causal Representation Meets Stochastic Modeling under Generic Geometry](https://mlsanity.com/paper/2602.05033/) (2026)
- [A General Representation-Based Approach to Multi-Source Domain Adaptation](https://mlsanity.com/paper/2604.23790/) (2026)
- [Intervening to Learn and Compose Causally Disentangled Representations](https://mlsanity.com/paper/2507.04754/) (2025)
- [Beyond identifiability: Learning causal representations with few environments and finite samples](https://mlsanity.com/paper/2603.25796/) (2026)
- [Identifiability of Potentially Degenerate Gaussian Mixture Models With Piecewise Affine Mixing](https://mlsanity.com/paper/2604.13218/) (2026)
- [Incorporating Interventional Independence Improves Robustness against Interventional Distribution Shift](https://mlsanity.com/paper/2507.05412/) (2025)
- [Discrete Causal Representations from Heterogeneous Domains: A Bayesian Approach with Social Survey Applications](https://mlsanity.com/paper/2606.06288/) (2026)
- [Jigsaw-CRL: Recovering Global Latent Causal Order from Fragmented Multi-Client Interventions](https://mlsanity.com/paper/2608.28991/) (2026)

## Abstract (arXiv)

The task of inferring high-level causal variables from low-level observations, commonly referred to as causal representation learning, is fundamentally underconstrained. As such, recent works to address this problem focus on various assumptions that lead to identifiability of the underlying latent causal variables. A large corpus of these preceding approaches consider multi-environment data collected under different interventions on the causal model. What is common to virtually all of these works is the restrictive assumption that in each environment, only a single variable is intervened on. In this work, we relax this assumption and provide the first identifiability result for causal representation learning that allows for multiple variables to be targeted by an intervention within one environment. Our approach hinges on a general assumption on the coverage and diversity of interventions across environments, which also includes the shared assumption of single-node interventions of previous works. The main idea behind our approach is to exploit the trace that interventions leave on the variance of the ground truth causal variables and regularizing for a specific notion of sparsity with respect to this trace. In addition to and inspired by our theoretical contributions, we present a practical algorithm to learn causal representations from multi-node interventional data and provide empirical evidence that validates our identifiability results.