# DAGs with No Curl: An Efficient DAG Structure Learning Approach

arXiv: https://arxiv.org/abs/2106.07197 · HTML: https://mlsanity.com/paper/2106.07197/

## Key facts

- Cited by 53 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 62 references that are in the map.
- First posted on arXiv 14 Jun 2021.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [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)
- [CAM: Causal additive models, high-dimensional order search and penalized regression](https://mlsanity.com/paper/1310.1533/) (2013)
- [Causal Discovery with Reinforcement Learning](https://mlsanity.com/paper/1906.04477/) (2019)
- [Masked Gradient-Based Causal Structure Learning](https://mlsanity.com/paper/1910.08527/) (2019)
- [DAGs with No Fears: A Closer Look at Continuous Optimization for Learning Bayesian Networks](https://mlsanity.com/paper/2010.09133/) (2020)
- [On Causal Discovery with Equal Variance Assumption](https://mlsanity.com/paper/1807.03419/) (2018)
- [On Iterative Hard Thresholding Methods for High-dimensional M-Estimation](https://mlsanity.com/paper/1410.5137/) (2014)
- [Being Bayesian about Network Structure](https://mlsanity.com/paper/1301.3856/) (2013)
- [Advances in Learning Bayesian Networks of Bounded Treewidth](https://mlsanity.com/paper/1406.1411/) (2014)
- [Learning DAGs without imposing acyclicity](https://mlsanity.com/paper/2006.03005/) (2020)
- [Structure Learning in Bayesian Networks of Moderate Size by Efficient Sampling](https://mlsanity.com/paper/1501.04370/) (2015)

## Built on by (most cited first, 30 of 53)

- [DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization](https://mlsanity.com/paper/2209.08037/) (2022)
- [Differentiable DAG Sampling](https://mlsanity.com/paper/2203.08509/) (2022)
- [Structure Learning with Continuous Optimization: A Sober Look and Beyond](https://mlsanity.com/paper/2304.02146/) (2023)
- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [A Survey on Causal Discovery Methods for I.I.D. and Time Series Data](https://mlsanity.com/paper/2303.15027/) (2023)
- [BayesDAG: Gradient-Based Posterior Inference for Causal Discovery](https://mlsanity.com/paper/2307.13917/) (2023)
- [Towards Federated Bayesian Network Structure Learning with Continuous Optimization](https://mlsanity.com/paper/2110.09356/) (2021)
- [Stable Differentiable Causal Discovery](https://mlsanity.com/paper/2311.10263/) (2023)
- [Truncated Matrix Power Iteration for Differentiable DAG Learning](https://mlsanity.com/paper/2208.14571/) (2022)
- [textttcausalAssembly: Generating Realistic Production Data for Benchmarking Causal Discovery](https://mlsanity.com/paper/2306.10816/) (2023)
- [FedDAG: Federated DAG Structure Learning](https://mlsanity.com/paper/2112.03555/) (2021)
- [Federated Causal Discovery from Heterogeneous Data](https://mlsanity.com/paper/2402.13241/) (2024)
- [Deep Causal Learning: Representation, Discovery and Inference](https://mlsanity.com/paper/2211.03374/) (2022)
- [Towards Privacy-Aware Causal Structure Learning in Federated Setting](https://mlsanity.com/paper/2211.06919/) (2022)
- [Constraint-Free Structure Learning with Smooth Acyclic Orientations](https://mlsanity.com/paper/2309.08406/) (2023)
- [Learning DAGs from Data with Few Root Causes](https://mlsanity.com/paper/2305.15936/) (2023)
- [Continuous Bayesian Model Selection for Multivariate Causal Discovery](https://mlsanity.com/paper/2411.10154/) (2024)
- [Modeling Time-evolving Causality over Data Streams](https://mlsanity.com/paper/2502.08963/) (2025)
- [Learning Latent Structural Causal Models](https://mlsanity.com/paper/2210.13583/) (2022)
- [Boosting Differentiable Causal Discovery via Adaptive Sample Reweighting](https://mlsanity.com/paper/2303.03187/) (2023)
- [Analytic DAG Constraints for Differentiable DAG Learning](https://mlsanity.com/paper/2503.19218/) (2025)
- [Discovering Dynamic Causal Space for DAG Structure Learning](https://mlsanity.com/paper/2306.02822/) (2023)
- [ProDAG: Projected Variational Inference for Directed Acyclic Graphs](https://mlsanity.com/paper/2405.15167/) (2024)
- [Since Faithfulness Fails: The Performance Limits of Neural Causal Discovery](https://mlsanity.com/paper/2502.16056/) (2025)
- [Learning Causal Graphs at Scale: A Foundation Model Approach](https://mlsanity.com/paper/2506.18285/) (2025)
- [The Robustness of Differentiable Causal Discovery in Misspecified Scenarios](https://mlsanity.com/paper/2510.12503/) (2025)
- [Learning Discrete Directed Acyclic Graphs via Backpropagation](https://mlsanity.com/paper/2210.15353/) (2022)
- [Reinforcement Learning for Causal Discovery without Acyclicity Constraints](https://mlsanity.com/paper/2408.13448/) (2024)
- [DAG-WGAN: Causal Structure Learning With Wasserstein Generative Adversarial Networks](https://mlsanity.com/paper/2204.00387/) (2022)
- [Directed Acyclic Graphs With Tears](https://mlsanity.com/paper/2302.02160/) (2023)

## Abstract (arXiv)

Recently directed acyclic graph (DAG) structure learning is formulated as a constrained continuous optimization problem with continuous acyclicity constraints and was solved iteratively through subproblem optimization. To further improve efficiency, we propose a novel learning framework to model and learn the weighted adjacency matrices in the DAG space directly. Specifically, we first show that the set of weighted adjacency matrices of DAGs are equivalent to the set of weighted gradients of graph potential functions, and one may perform structure learning by searching in this equivalent set of DAGs. To instantiate this idea, we propose a new algorithm, DAG-NoCurl, which solves the optimization problem efficiently with a two-step procedure: 1) first we find an initial cyclic solution to the optimization problem, and 2) then we employ the Hodge decomposition of graphs and learn an acyclic graph by projecting the cyclic graph to the gradient of a potential function. Experimental studies on benchmark datasets demonstrate that our method provides comparable accuracy but better efficiency than baseline DAG structure learning methods on both linear and generalized structural equation models, often by more than one order of magnitude.