# Score-based Causal Representation Learning: Linear and General Transformations

arXiv: https://arxiv.org/abs/2402.00849 · HTML: https://mlsanity.com/paper/2402.00849/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Feb 2024.

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

- [Linear Causal Representation Learning from Unknown Multi-node Interventions](https://mlsanity.com/paper/2406.05937/) (2024)
- [Causality is Key for Interpretability Claims to Generalise](https://mlsanity.com/paper/2602.16698/) (2026)
- [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)
- [Towards Identifiable Latent Additive Noise Models](https://mlsanity.com/paper/2403.15711/) (2024)
- [Causal Representation Learning from General Environments under Nonparametric Mixing](https://mlsanity.com/paper/2604.23800/) (2026)
- [Revealing Multimodal Causality with Large Language Models](https://mlsanity.com/paper/2509.17784/) (2025)
- [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)
- [A Conditional Distribution Equality Testing Framework using Deep Generative Learning](https://mlsanity.com/paper/2509.17729/) (2025)
- [Multi-Domain Empirical Bayes for Linearly-Mixed Causal Representations](https://mlsanity.com/paper/2603.18404/) (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)
- [Blind Thermodynamic Ontology Discovery from Anonymous Experiments](https://mlsanity.com/paper/2609.23387/) (2026)
- [Extrapolation Guarantees for Perturbation Modeling Under the Additive Latent Shift Assumption](https://mlsanity.com/paper/2504.18522/) (2025)
- [Linear Causal Representation Learning by Topological Ordering, Pruning, and Disentanglement](https://mlsanity.com/paper/2509.22553/) (2025)
- [On the Identifiability of Causal Graphs with the Invariance Principle](https://mlsanity.com/paper/2510.13583/) (2025)
- [ROPES: Robotic Pose Estimation via Score-Based Causal Representation Learning](https://mlsanity.com/paper/2510.20884/) (2025)
- [Eigenfunction Extraction for Ordered Representation Learning](https://mlsanity.com/paper/2510.24672/) (2025)
- [Proxy-Guided Measurement Calibration](https://mlsanity.com/paper/2603.09288/) (2026)
- [From Generalist to Specialist Representation](https://mlsanity.com/paper/2605.12733/) (2026)
- [Causal Learning with the Invariance Principle](https://mlsanity.com/paper/2605.13589/) (2026)
- [A Dialogue between Causal and Traditional Representation Learning: Toward Mutual Benefits in a Unified Formulation](https://mlsanity.com/paper/2605.21058/) (2026)
- [Discrete Causal Representations from Heterogeneous Domains: A Bayesian Approach with Social Survey Applications](https://mlsanity.com/paper/2606.06288/) (2026)
- [Conditionally Identifiable Latent-Environment Modeling for Out-of-Distribution Recommendation](https://mlsanity.com/paper/2608.03647/) (2026)
- [Jigsaw-CRL: Recovering Global Latent Causal Order from Fragmented Multi-Client Interventions](https://mlsanity.com/paper/2608.28991/) (2026)
- [Causal Retention in Interactive Agents: Interface Factorization and Selective Adaptation](https://mlsanity.com/paper/2609.30650/) (2026)

## Abstract (arXiv)

This paper addresses intervention-based causal representation learning (CRL) under a general nonparametric latent causal model and an unknown transformation that maps the latent variables to the observed variables. Linear and general transformations are investigated. The paper addresses both the identifiability and achievability aspects. Identifiability refers to determining algorithm-agnostic conditions that ensure the recovery of the true latent causal variables and the underlying latent causal graph. Achievability refers to the algorithmic aspects and addresses designing algorithms that achieve identifiability guarantees. By drawing novel connections between score functions (i.e., the gradients of the logarithm of density functions) and CRL, this paper designs a score-based class of algorithms that ensures both identifiability and achievability. First, the paper focuses on linear transformations and shows that one stochastic hard intervention per node suffices to guarantee identifiability. It also provides partial identifiability guarantees for soft interventions, including identifiability up to mixing with parents for general causal models and perfect recovery of the latent graph for sufficiently nonlinear causal models. Secondly, it focuses on general transformations and demonstrates that two stochastic hard interventions per node are sufficient for identifiability. This is achieved by defining a differentiable loss function whose global optima ensure identifiability for general CRL. Notably, one does not need to know which pair of interventional environments has the same node intervened. Finally, the theoretical results are empirically validated via experiments on structured synthetic data and image data.