# NODAGS-Flow: Nonlinear Cyclic Causal Structure Learning

arXiv: https://arxiv.org/abs/2301.01849 · HTML: https://mlsanity.com/paper/2301.01849/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 54 references that are in the map.
- First posted on arXiv 4 Jan 2023.

## Builds on (most cited first, 24 of 24)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [DAG-GNN: DAG Structure Learning with Graph Neural Networks](https://mlsanity.com/paper/1904.10098/) (2019)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [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)
- [Invariant Causal Prediction for Nonlinear Models](https://mlsanity.com/paper/1706.08576/) (2017)
- [Causal Autoregressive Flows](https://mlsanity.com/paper/2011.02268/) (2020)
- [Masked Gradient-Based Causal Structure Learning](https://mlsanity.com/paper/1910.08527/) (2019)
- [Permutation-based Causal Inference Algorithms with Interventions](https://mlsanity.com/paper/1705.10220/) (2017)
- [Constraint-based Causal Discovery from Multiple Interventions over Overlapping Variable Sets](https://mlsanity.com/paper/1403.2150/) (2014)
- [Structural Agnostic Modeling: Adversarial Learning of Causal Graphs](https://mlsanity.com/paper/1803.04929/) (2018)
- [Large-Scale Differentiable Causal Discovery of Factor Graphs](https://mlsanity.com/paper/2206.07824/) (2022)
- [Scaling structural learning with NO-BEARS to infer causal transcriptome networks](https://mlsanity.com/paper/1911.00081/) (2019)
- [A Discovery Algorithm for Directed Cyclic Graphs](https://mlsanity.com/paper/1302.3599/) (2013)
- [The Case for Evaluating Causal Models Using Interventional Measures and Empirical Data](https://mlsanity.com/paper/1910.05387/) (2019)
- [Cyclic Causal Discovery from Continuous Equilibrium Data](https://mlsanity.com/paper/1309.6849/) (2013)
- [Structure Learning for Cyclic Linear Causal Models](https://mlsanity.com/paper/2006.05978/) (2020)

## Built on by (most cited first, 20 of 20)

- [GFlowNets for AI-Driven Scientific Discovery](https://mlsanity.com/paper/2302.00615/) (2023)
- [Joint Bayesian Inference of Graphical Structure and Parameters with a Single Generative Flow Network](https://mlsanity.com/paper/2305.19366/) (2023)
- [Causal Modeling with Stationary Diffusions](https://mlsanity.com/paper/2310.17405/) (2023)
- [The Landscape of Causal Discovery Data: Grounding Causal Discovery in Real-World Applications](https://mlsanity.com/paper/2412.01953/) (2024)
- [Recursive Causal Discovery](https://mlsanity.com/paper/2403.09300/) (2024)
- [Scalable and Flexible Causal Discovery with an Efficient Test for Adjacency](https://mlsanity.com/paper/2406.09177/) (2024)
- [Differentiable Cyclic Causal Discovery Under Unmeasured Confounders](https://mlsanity.com/paper/2508.08450/) (2025)
- [Differentiable Causal Discovery For Latent Hierarchical Causal Models](https://mlsanity.com/paper/2411.19556/) (2024)
- [Score-based Greedy Search for Structure Identification of Partially Observed Linear Causal Models](https://mlsanity.com/paper/2510.04378/) (2025)
- [Efficient Learning of Stationary Diffusions with Stein-type Discrepancies](https://mlsanity.com/paper/2601.16597/) (2026)
- [Learning Cyclic Causal Models from Incomplete Data](https://mlsanity.com/paper/2402.15625/) (2024)
- [MissNODAG: Differentiable Cyclic Causal Graph Learning from Incomplete Data](https://mlsanity.com/paper/2410.18918/) (2024)
- [Extended Fiducial Inference for Individual Treatment Effects via Deep Neural Networks](https://mlsanity.com/paper/2505.01995/) (2025)
- [Consistency evaluation of benchmarks used for causal discovery](https://mlsanity.com/paper/2606.01789/) (2026)
- [Partial Causal Structure Learning for Valid Selective Conformal Inference under Interventions](https://mlsanity.com/paper/2603.02204/) (2026)
- [RECLAIM: Cyclic Causal Discovery Amid Measurement Noise](https://mlsanity.com/paper/2603.20585/) (2026)
- [SCOUT: Cyclic Causal Discovery Under Soft Interventions with Unknown Targets](https://mlsanity.com/paper/2605.16620/) (2026)
- [SVI-DAG: A Structured Variational Inference Approach to Bayesian Causal Discovery](https://mlsanity.com/paper/2608.04930/) (2026)
- [CausalArena: Benchmarking Causal Discovery in the Foundation Model Era](https://mlsanity.com/paper/2609.11897/) (2026)
- [One Intervention per Component is Enough: Towards Identifiability in Linear Stochastic Dynamics from Steady State](https://mlsanity.com/paper/2609.19955/) (2026)

## Abstract (arXiv)

Learning causal relationships between variables is a well-studied problem in statistics, with many important applications in science. However, modeling real-world systems remain challenging, as most existing algorithms assume that the underlying causal graph is acyclic. While this is a convenient framework for developing theoretical developments about causal reasoning and inference, the underlying modeling assumption is likely to be violated in real systems, because feedback loops are common (e.g., in biological systems). Although a few methods search for cyclic causal models, they usually rely on some form of linearity, which is also limiting, or lack a clear underlying probabilistic model. In this work, we propose a novel framework for learning nonlinear cyclic causal graphical models from interventional data, called NODAGS-Flow. We perform inference via direct likelihood optimization, employing techniques from residual normalizing flows for likelihood estimation. Through synthetic experiments and an application to single-cell high-content perturbation screening data, we show significant performance improvements with our approach compared to state-of-the-art methods with respect to structure recovery and predictive performance.