# Effective Causal Discovery under Identifiable Heteroscedastic Noise Model

arXiv: https://arxiv.org/abs/2312.12844 · HTML: https://mlsanity.com/paper/2312.12844/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 40 references that are in the map.
- First posted on arXiv 20 Dec 2023.

## Builds on (most cited first, 22 of 22)

- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [DAG-GNN: DAG Structure Learning with Graph Neural Networks](https://mlsanity.com/paper/1904.10098/) (2019)
- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [Causality for Machine Learning](https://mlsanity.com/paper/1911.10500/) (2019)
- [Gradient-Based Neural DAG Learning](https://mlsanity.com/paper/1906.02226/) (2019)
- [Learning Sparse Nonparametric DAGs](https://mlsanity.com/paper/1909.13189/) (2019)
- [On the Role of Sparsity and DAG Constraints for Learning Linear DAGs](https://mlsanity.com/paper/2006.10201/) (2020)
- [Causal Inference in the Presence of Latent Variables and Selection Bias](https://mlsanity.com/paper/1302.4983/) (2013)
- [Causal Discovery from Heterogeneous/Nonstationary Data with Independent Changes](https://mlsanity.com/paper/1903.01672/) (2019)
- [Beware of the Simulated DAG! Causal Discovery Benchmarks May Be Easy To Game](https://mlsanity.com/paper/2102.13647/) (2021)
- [Causal Autoregressive Flows](https://mlsanity.com/paper/2011.02268/) (2020)
- [Masked Gradient-Based Causal Structure Learning](https://mlsanity.com/paper/1910.08527/) (2019)
- [On Causal Discovery with Equal Variance Assumption](https://mlsanity.com/paper/1807.03419/) (2018)
- [On the Identifiability and Estimation of Causal Location-Scale Noise Models](https://mlsanity.com/paper/2210.09054/) (2022)
- [On the Convergence of Continuous Constrained Optimization for Structure Learning](https://mlsanity.com/paper/2011.11150/) (2020)
- [A polynomial-time algorithm for learning nonparametric causal graphs](https://mlsanity.com/paper/2006.11970/) (2020)
- [Causally Regularized Learning with Agnostic Data Selection Bias](https://mlsanity.com/paper/1708.06656/) (2017)
- [Detecting non-causal artifacts in multivariate linear regression models](https://mlsanity.com/paper/1803.00810/) (2018)
- [Causal Discovery with Heterogeneous Observational Data](https://mlsanity.com/paper/2201.12392/) (2022)
- [Structure learning in polynomial time: Greedy algorithms, Bregman information, and exponential families](https://mlsanity.com/paper/2110.04719/) (2021)
- [Distinguishing Cause from Effect Based on Exogeneity](https://mlsanity.com/paper/1504.05651/) (2015)
- [Scalable Causal Domain Adaptation](https://mlsanity.com/paper/2103.00139/) (2021)

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

- [A Skewness-Based Criterion for Addressing Heteroscedastic Noise in Causal Discovery](https://mlsanity.com/paper/2410.06407/) (2024)
- [Learning Causal Graphs at Scale: A Foundation Model Approach](https://mlsanity.com/paper/2506.18285/) (2025)
- [CDFM: Towards a General-Purpose Causal Discovery Foundation Model](https://mlsanity.com/paper/2607.11508/) (2026)
- [DAG-FM: A Foundation Model for Causal Discovery under Heterogeneous Causal Mechanisms](https://mlsanity.com/paper/2607.11510/) (2026)
- [Consistency evaluation of benchmarks used for causal discovery](https://mlsanity.com/paper/2606.01789/) (2026)
- [Beyond Additivity: Causal Discovery in Location-Scale Noise Models with Hidden Variables](https://mlsanity.com/paper/2606.08196/) (2026)
- [Moment Matters: Mean and Variance Causal Graph Discovery from Heteroscedastic Observational Data](https://mlsanity.com/paper/2602.23602/) (2026)
- [Auditing Marketing Budget Allocation with Hindsight Regret](https://mlsanity.com/paper/2604.25977/) (2026)
- [Intervention-Based Time Series Causal Discovery via Simulator-Generated Interventional Distributions](https://mlsanity.com/paper/2605.09870/) (2026)
- [DAG Structure Learning with Node-Varying Heavy-Tailed Disturbances](https://mlsanity.com/paper/2610.13304/) (2026)

## Abstract (arXiv)

Capturing the underlying structural causal relations represented by Directed Acyclic Graphs (DAGs) has been a fundamental task in various AI disciplines. Causal DAG learning via the continuous optimization framework has recently achieved promising performance in terms of both accuracy and efficiency. However, most methods make strong assumptions of homoscedastic noise, i.e., exogenous noises have equal variances across variables, observations, or even both. The noises in real data usually violate both assumptions due to the biases introduced by different data collection processes. To address the issue of heteroscedastic noise, we introduce relaxed and implementable sufficient conditions, proving the identifiability of a general class of SEM subject to these conditions. Based on the identifiable general SEM, we propose a novel formulation for DAG learning that accounts for the variation in noise variance across variables and observations. We then propose an effective two-phase iterative DAG learning algorithm to address the increasing optimization difficulties and to learn a causal DAG from data with heteroscedastic variable noise under varying variance. We show significant empirical gains of the proposed approaches over state-of-the-art methods on both synthetic data and real data.