# Causal Discovery in Linear Structural Causal Models with Deterministic Relations

arXiv: https://arxiv.org/abs/2111.00341 · HTML: https://mlsanity.com/paper/2111.00341/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 52 references that are in the map.
- First posted on arXiv 30 Oct 2021.

## Builds on (most cited first, 5 of 5)

- [Learning Linear Non-Gaussian Causal Models in the Presence of Latent Variables](https://mlsanity.com/paper/1908.03932/) (2019)
- [ParceLiNGAM: A causal ordering method robust against latent confounders](https://mlsanity.com/paper/1303.7410/) (2013)
- [Causal Discovery with Unobserved Confounding and non-Gaussian Data](https://mlsanity.com/paper/2007.11131/) (2020)
- [Analysis and Control of Epidemics: A survey of spreading processes on complex networks](https://mlsanity.com/paper/1505.00768/) (2015)
- [Multiscale Blind Source Separation](https://mlsanity.com/paper/1608.07173/) (2016)

## Built on by (most cited first, 4 of 4)

- [Causal Discovery in Linear Latent Variable Models Subject to Measurement Error](https://mlsanity.com/paper/2211.03984/) (2022)
- [Learning Unknown Intervention Targets in Structural Causal Models from Heterogeneous Data](https://mlsanity.com/paper/2312.06091/) (2023)
- [Arrow: A Foundation Model for Causal Discovery](https://mlsanity.com/paper/2605.07204/) (2026)
- [Relaxed Sparsest-Permutation Formulation for Causal Discovery at Scale](https://mlsanity.com/paper/2605.05568/) (2026)

## Abstract (arXiv)

Linear structural causal models (SCMs) -- in which each observed variable is generated by a subset of the other observed variables as well as a subset of the exogenous sources -- are pervasive in causal inference and casual discovery. However, for the task of causal discovery, existing work almost exclusively focus on the submodel where each observed variable is associated with a distinct source with non-zero variance. This results in the restriction that no observed variable can deterministically depend on other observed variables or latent confounders. In this paper, we extend the results on structure learning by focusing on a subclass of linear SCMs which do not have this property, i.e., models in which observed variables can be causally affected by any subset of the sources, and are allowed to be a deterministic function of other observed variables or latent confounders. This allows for a more realistic modeling of influence or information propagation in systems. We focus on the task of causal discovery form observational data generated from a member of this subclass. We derive a set of necessary and sufficient conditions for unique identifiability of the causal structure. To the best of our knowledge, this is the first work that gives identifiability results for causal discovery under both latent confounding and deterministic relationships. Further, we propose an algorithm for recovering the underlying causal structure when the aforementioned conditions are satisfied. We validate our theoretical results both on synthetic and real datasets.