# DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization

arXiv: https://arxiv.org/abs/2209.08037 · HTML: https://mlsanity.com/paper/2209.08037/

## Key facts

- Cited by 140 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 72 references that are in the map.
- First posted on arXiv 16 Sep 2022.

## Builds on (most cited first, 27 of 27)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [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)
- [DYNOTEARS: Structure Learning from Time-Series Data](https://mlsanity.com/paper/2002.00498/) (2020)
- [On the Role of Sparsity and DAG Constraints for Learning Linear DAGs](https://mlsanity.com/paper/2006.10201/) (2020)
- [Learning Bayesian Networks: The Combination of Knowledge and Statistical Data](https://mlsanity.com/paper/1302.6815/) (2013)
- [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)
- [High-dimensional learning of linear causal networks via inverse covariance estimation](https://mlsanity.com/paper/1311.3492/) (2013)
- [DAGs with No Fears: A Closer Look at Continuous Optimization for Learning Bayesian Networks](https://mlsanity.com/paper/2010.09133/) (2020)
- [Causal Discovery with General Non-Linear Relationships Using Non-Linear ICA](https://mlsanity.com/paper/1904.09096/) (2019)
- [Differentiable Causal Discovery Under Unmeasured Confounding](https://mlsanity.com/paper/2010.06978/) (2020)
- [DAGs with No Curl: An Efficient DAG Structure Learning Approach](https://mlsanity.com/paper/2106.07197/) (2021)
- [Learning linear structural equation models in polynomial time and sample complexity](https://mlsanity.com/paper/1707.04673/) (2017)
- [CASTLE: Regularization via Auxiliary Causal Graph Discovery](https://mlsanity.com/paper/2009.13180/) (2020)
- [On the Convergence of Continuous Constrained Optimization for Structure Learning](https://mlsanity.com/paper/2011.11150/) (2020)
- [Concave Penalized Estimation of Sparse Gaussian Bayesian Networks](https://mlsanity.com/paper/1401.0852/) (2014)
- [Learning Identifiable Gaussian Bayesian Networks in Polynomial Time and Sample Complexity](https://mlsanity.com/paper/1703.01196/) (2017)
- [Learning Quadratic Variance Function (QVF) DAG models via OverDispersion Scoring (ODS)](https://mlsanity.com/paper/1704.08783/) (2017)
- [Causal Adversarial Network for Learning Conditional and Interventional Distributions](https://mlsanity.com/paper/2008.11376/) (2020)
- [Penalized Estimation of Directed Acyclic Graphs From Discrete Data](https://mlsanity.com/paper/1403.2310/) (2014)
- [High-Dimensional Poisson DAG Model Learning Using ℓ1-Regularized Regression](https://mlsanity.com/paper/1810.02501/) (2018)
- [Graph estimation with joint additive models](https://mlsanity.com/paper/1304.4654/) (2013)

## Built on by (most cited first, 30 of 140)

- [Learning Linear Causal Representations from Interventions under General Nonlinear Mixing](https://mlsanity.com/paper/2306.02235/) (2023)
- [Fast Scalable and Accurate Discovery of DAGs Using the Best Order Score Search and Grow-Shrink Trees](https://mlsanity.com/paper/2310.17679/) (2023)
- [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)
- [Stable Differentiable Causal Discovery](https://mlsanity.com/paper/2311.10263/) (2023)
- [Reimagining Synthetic Tabular Data Generation through Data-Centric AI: A Comprehensive Benchmark](https://mlsanity.com/paper/2310.16981/) (2023)
- [Optimizing NOTEARS Objectives via Topological Swaps](https://mlsanity.com/paper/2305.17277/) (2023)
- [ACE : Off-Policy Actor-Critic with Causality-Aware Entropy Regularization](https://mlsanity.com/paper/2402.14528/) (2024)
- [CoLiDE: Concomitant Linear DAG Estimation](https://mlsanity.com/paper/2310.02895/) (2023)
- [Ordering-Based Causal Discovery for Linear and Nonlinear Relations](https://mlsanity.com/paper/2410.05890/) (2024)
- [Demystifying amortized causal discovery with transformers](https://mlsanity.com/paper/2405.16924/) (2024)
- [Global Optimality in Bivariate Gradient-based DAG Learning](https://mlsanity.com/paper/2306.17378/) (2023)
- [Constraint-Free Structure Learning with Smooth Acyclic Orientations](https://mlsanity.com/paper/2309.08406/) (2023)
- [DAG Learning on the Permutahedron](https://mlsanity.com/paper/2301.11898/) (2023)
- [Learning DAGs from Data with Few Root Causes](https://mlsanity.com/paper/2305.15936/) (2023)
- [DCILP: A Distributed Approach for Large-Scale Causal Structure Learning](https://mlsanity.com/paper/2406.10481/) (2024)
- [C2P: Featuring Large Language Models with Causal Reasoning](https://mlsanity.com/paper/2407.18069/) (2024)
- [Non-negative Weighted DAG Structure Learning](https://mlsanity.com/paper/2409.07880/) (2024)
- [Markov Equivalence and Consistency in Differentiable Structure Learning](https://mlsanity.com/paper/2410.06163/) (2024)
- [A Skewness-Based Criterion for Addressing Heteroscedastic Noise in Causal Discovery](https://mlsanity.com/paper/2410.06407/) (2024)
- [Predictive Coding beyond Correlations](https://mlsanity.com/paper/2306.15479/) (2023)
- [Dagma-DCE: Interpretable, Non-Parametric Differentiable Causal Discovery](https://mlsanity.com/paper/2401.02930/) (2024)
- [Optimal Transport for Structure Learning Under Missing Data](https://mlsanity.com/paper/2402.15255/) (2024)
- [Better Simulations for Validating Causal Discovery with the DAG-Adaptation of the Onion Method](https://mlsanity.com/paper/2405.13100/) (2024)
- [Scalable and Flexible Causal Discovery with an Efficient Test for Adjacency](https://mlsanity.com/paper/2406.09177/) (2024)
- [LLM-initialized Differentiable Causal Discovery](https://mlsanity.com/paper/2410.21141/) (2024)
- [Continuous Bayesian Model Selection for Multivariate Causal Discovery](https://mlsanity.com/paper/2411.10154/) (2024)
- [Boosting Differentiable Causal Discovery via Adaptive Sample Reweighting](https://mlsanity.com/paper/2303.03187/) (2023)
- [iSCAN: Identifying Causal Mechanism Shifts among Nonlinear Additive Noise Models](https://mlsanity.com/paper/2306.17361/) (2023)
- [Fairness-Driven LLM-based Causal Discovery with Active Learning and Dynamic Scoring](https://mlsanity.com/paper/2503.17569/) (2025)

## Abstract (arXiv)

The combinatorial problem of learning directed acyclic graphs (DAGs) from data was recently framed as a purely continuous optimization problem by leveraging a differentiable acyclicity characterization of DAGs based on the trace of a matrix exponential function. Existing acyclicity characterizations are based on the idea that powers of an adjacency matrix contain information about walks and cycles. In this work, we propose a new acyclicity characterization based on the log-determinant (log-det) function, which leverages the nilpotency property of DAGs. To deal with the inherent asymmetries of a DAG, we relate the domain of our log-det characterization to the set of textitM-matrices, which is a key difference to the classical log-det function defined over the cone of positive definite matrices. Similar to acyclicity functions previously proposed, our characterization is also exact and differentiable. However, when compared to existing characterizations, our log-det function: (1) Is better at detecting large cycles; (2) Has better-behaved gradients; and (3) Its runtime is in practice about an order of magnitude faster. From the optimization side, we drop the typically used augmented Lagrangian scheme and propose DAGMA (textitDAGs via M-matrices for Acyclicity), a method that resembles the central path for barrier methods. Each point in the central path of DAGMA is a solution to an unconstrained problem regularized by our log-det function, then we show that at the limit of the central path the solution is guaranteed to be a DAG. Finally, we provide extensive experiments for textitlinear and textitnonlinear SEMs and show that our approach can reach large speed-ups and smaller structural Hamming distances against state-of-the-art methods. Code implementing the proposed method is open-source and publicly available at https://github.com/kevinsbello/dagma.