mlsanityData to 14 Oct 2026

DAGs with No Curl: An Efficient DAG Structure Learning Approach

arXiv 2106.07197 ↗First posted 14 Jun 2021cs.LG cs.AI stat.MLcited by 53

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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.

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

17 of 62 references.

References · 17

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2018DAGs with NO TEARS: Continuous Optimization for Structure Learningcited by 752
  3. 2019DAG-GNN: DAG Structure Learning with Graph Neural Networkscited by 331
  4. 2013Causal Discovery with Continuous Additive Noise Modelscited by 255
  5. 2019Gradient-Based Neural DAG Learningcited by 227
  6. 2019Learning Sparse Nonparametric DAGscited by 207
  7. 2020On the Role of Sparsity and DAG Constraints for Learning Linear DAGscited by 191
  8. 2013CAM: Causal additive models, high-dimensional order search and penalized regressioncited by 178
  9. 2019Causal Discovery with Reinforcement Learningcited by 145
  10. 2019Masked Gradient-Based Causal Structure Learningcited by 89
  11. 2020DAGs with No Fears: A Closer Look at Continuous Optimization for Learning Bayesian Networkscited by 66
  12. 2018On Causal Discovery with Equal Variance Assumptioncited by 59
5 more
  1. 2014On Iterative Hard Thresholding Methods for High-dimensional M-Estimationcited by 53
  2. 2013Being Bayesian about Network Structurecited by 17
  3. 2014Advances in Learning Bayesian Networks of Bounded Treewidthcited by 10
  4. 2020Learning DAGs without imposing acyclicitycited by 10
  5. 2015Structure Learning in Bayesian Networks of Moderate Size by Efficient Samplingcited by 3

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 3; 2022: 10; 2023: 13; 2024: 8; 2025: 9; 2026: 10.

53 papers.

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  1. 2022DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterizationcited by 140
  2. 2022Differentiable DAG Samplingcited by 47
  3. 2023Structure Learning with Continuous Optimization: A Sober Look and Beyondcited by 36
  4. 2023Discovering Causal Relations and Equations from Datacited by 36
  5. 2023A Survey on Causal Discovery Methods for I.I.D. and Time Series Datacited by 35
  6. 2023BayesDAG: Gradient-Based Posterior Inference for Causal Discoverycited by 33
  7. 2021Towards Federated Bayesian Network Structure Learning with Continuous Optimizationcited by 31
  8. 2023Stable Differentiable Causal Discoverycited by 31
  9. 2022Truncated Matrix Power Iteration for Differentiable DAG Learningcited by 23
  10. 2023textttcausalAssembly: Generating Realistic Production Data for Benchmarking Causal Discoverycited by 18
  11. 2021FedDAG: Federated DAG Structure Learningcited by 17
  12. 2024Federated Causal Discovery from Heterogeneous Datacited by 15
38 more
  1. 2022Deep Causal Learning: Representation, Discovery and Inferencecited by 11
  2. 2022Towards Privacy-Aware Causal Structure Learning in Federated Settingcited by 11
  3. 2023Constraint-Free Structure Learning with Smooth Acyclic Orientationscited by 10
  4. 2023Learning DAGs from Data with Few Root Causescited by 8
  5. 2024Continuous Bayesian Model Selection for Multivariate Causal Discoverycited by 6
  6. 2025Modeling Time-evolving Causality over Data Streamscited by 6
  7. 2022Learning Latent Structural Causal Modelscited by 5
  8. 2023Boosting Differentiable Causal Discovery via Adaptive Sample Reweightingcited by 5
  9. 2025Analytic DAG Constraints for Differentiable DAG Learningcited by 5
  10. 2023Discovering Dynamic Causal Space for DAG Structure Learningcited by 4
  11. 2024ProDAG: Projected Variational Inference for Directed Acyclic Graphscited by 4
  12. 2025Since Faithfulness Fails: The Performance Limits of Neural Causal Discoverycited by 4
  13. 2025Learning Causal Graphs at Scale: A Foundation Model Approachcited by 4
  14. 2025The Robustness of Differentiable Causal Discovery in Misspecified Scenarioscited by 4
  15. 2022Learning Discrete Directed Acyclic Graphs via Backpropagationcited by 3
  16. 2024Reinforcement Learning for Causal Discovery without Acyclicity Constraintscited by 3
  17. 2022DAG-WGAN: Causal Structure Learning With Wasserstein Generative Adversarial Networkscited by 2
  18. 2023Directed Acyclic Graphs With Tearscited by 2
  19. 2024LOCAL: Learning with Orientation Matrix to Infer Causal Structure from Time Series Datacited by 2
  20. 2025Robust Causal Discovery under Imperfect Structural Constraintscited by 2
  21. 2021A Stochastic Variance-Reduced Coordinate Descent Algorithm for Learning Sparse Bayesian Network from Discrete High-Dimensional Datacited by 1
  22. 2023Causal disentanglement of multimodal datacited by 1
  23. 2024ExDAG: an MIQP Algorithm for Learning DAGscited by 1
  24. 2024Scalable Variational Causal Discovery Unconstrained by Acyclicitycited by 1
  25. 2025Causal Discovery via Bayesian Optimizationcited by 1
  26. 2026KG-SoftMAP: Soft Knowledge-Graph Priors for Bayesian Network Structure Learning from Sparse Discrete Datacited by 1
  27. 2022Causality Learning With Wasserstein Generative Adversarial Networkscited by 0
  28. 2022Differentiable and Transportable Structure Learningcited by 0
  29. 2023Distributionally Robust Skeleton Learning of Discrete Bayesian Networkscited by 0
  30. 2024Learning Dynamic Bayesian Networks from Data: Foundations, First Principles and Numerical Comparisonscited by 0
  31. 2025Sequential Causal Discovery with Noisy Language Model Priorscited by 0
  32. 2025PersonaX: Multimodal Datasets with LLM-Inferred Behavior Traitscited by 0
  33. 2026SC3D: Dynamic and Differentiable Causal Discovery for Temporal and Instantaneous Graphscited by 0
  34. 2026Smoothing the Landscape: Causal Structure Learning via Diffusion Denoising Objectivescited by 0
  35. 2026Certified Policy Optimisation for Nested Causal Bandits via PAC-Bayes Riskcited by 0
  36. 2026Learning Temporal Causal Structure via Smooth Differentiable Optimizationcited by 0
  37. 2026Causal Atlases from Entropic Inference: Bayesian Networks beyond Optimal DAGscited by 0
  38. 2026Amortized Bayesian Causal Discovery of Extended Factor Graphscited by 0

50 most cited of 53 All 53 (JSON)