# Score matching through the roof: linear, nonlinear, and latent variables causal discovery

arXiv: https://arxiv.org/abs/2407.18755 · HTML: https://mlsanity.com/paper/2407.18755/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 36 references that are in the map.
- First posted on arXiv 26 Jul 2024.

## Builds on (most cited first, 16 of 16)

- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [CAM: Causal additive models, high-dimensional order search and penalized regression](https://mlsanity.com/paper/1310.1533/) (2013)
- [Beware of the Simulated DAG! Causal Discovery Benchmarks May Be Easy To Game](https://mlsanity.com/paper/2102.13647/) (2021)
- [Score matching enables causal discovery of nonlinear additive noise models](https://mlsanity.com/paper/2203.04413/) (2022)
- [Efficient Neural Causal Discovery without Acyclicity Constraints](https://mlsanity.com/paper/2107.10483/) (2021)
- [Learning to Induce Causal Structure](https://mlsanity.com/paper/2204.04875/) (2022)
- [Diffusion Models for Causal Discovery via Topological Ordering](https://mlsanity.com/paper/2210.06201/) (2022)
- [Gradient Estimators for Implicit Models](https://mlsanity.com/paper/1705.07107/) (2017)
- [Learning linear structural equation models in polynomial time and sample complexity](https://mlsanity.com/paper/1707.04673/) (2017)
- [Scalable Causal Discovery with Score Matching](https://mlsanity.com/paper/2304.03382/) (2023)
- [Inference via low-dimensional couplings](https://mlsanity.com/paper/1703.06131/) (2017)
- [Causal Discovery with Score Matching on Additive Models with Arbitrary Noise](https://mlsanity.com/paper/2304.03265/) (2023)
- [Assumption violations in causal discovery and the robustness of score matching](https://mlsanity.com/paper/2310.13387/) (2023)
- [Sample Complexity Bounds for Score-Matching: Causal Discovery and Generative Modeling](https://mlsanity.com/paper/2310.18123/) (2023)
- [Shortcuts for causal discovery of nonlinear models by score matching](https://mlsanity.com/paper/2310.14246/) (2023)

## Built on by (most cited first, 10 of 10)

- [LoSAM: Local Search in Additive Noise Models with Mixed Mechanisms and General Noise for Global Causal Discovery](https://mlsanity.com/paper/2410.11759/) (2024)
- [When Additive Noise Meets Unobserved Mediators: Bivariate Denoising Diffusion for Causal Discovery](https://mlsanity.com/paper/2506.23374/) (2025)
- [Ordering-based Causal Discovery via Generalized Score Matching](https://mlsanity.com/paper/2601.16249/) (2026)
- [Order-based Causal Discovery for Multistage Processes](https://mlsanity.com/paper/2607.03971/) (2026)
- [Estimating Treatment Effects with Independent Component Analysis](https://mlsanity.com/paper/2507.16467/) (2025)
- [On the Identifiability of Causal Graphs with the Invariance Principle](https://mlsanity.com/paper/2510.13583/) (2025)
- [Position: Explainable AI is Causality in Disguise](https://mlsanity.com/paper/2603.28597/) (2026)
- [Causal Learning with the Invariance Principle](https://mlsanity.com/paper/2605.13589/) (2026)
- [Local Fisher Information Enables Sparse Causal Discovery](https://mlsanity.com/paper/2610.04291/) (2026)
- [Causal Discovery Under Hard Selection Bias: A New Robust Score-Matching Approach](https://mlsanity.com/paper/2610.18075/) (2026)

## Abstract (arXiv)

Causal discovery from observational data holds great promise, but existing methods rely on strong assumptions about the underlying causal structure, often requiring full observability of all relevant variables. We tackle these challenges by leveraging the score function nabla log p(X) of observed variables for causal discovery and propose the following contributions. First, we fine-tune the existing identifiability results with the score on additive noise models, showing that their assumption of nonlinearity of the causal mechanisms is not necessary. Second, we establish conditions for inferring causal relations from the score even in the presence of hidden variables; this result is two-faced: we demonstrate the score's potential to infer the equivalence class of causal graphs with hidden variables (while previous results are restricted to the fully observable setting), and we provide sufficient conditions for identifying direct causes in latent variable models. Building on these insights, we propose a flexible algorithm suited for causal discovery on linear, nonlinear, and latent variable models, which we empirically validate.