# Concomitant DAG Learning: On the Roles of Noise Adaptivity, Sparsity, and Non-negativity

arXiv: https://arxiv.org/abs/2605.23537 · HTML: https://mlsanity.com/paper/2605.23537/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 44 references that are in the map.
- First posted on arXiv 22 May 2026.

## 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)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [D'ya like DAGs? A Survey on Structure Learning and Causal Discovery](https://mlsanity.com/paper/2103.02582/) (2021)
- [On the Role of Sparsity and DAG Constraints for Learning Linear DAGs](https://mlsanity.com/paper/2006.10201/) (2020)
- [Beware of the Simulated DAG! Causal Discovery Benchmarks May Be Easy To Game](https://mlsanity.com/paper/2102.13647/) (2021)
- [DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization](https://mlsanity.com/paper/2209.08037/) (2022)
- [Connecting the Dots: Identifying Network Structure via Graph Signal Processing](https://mlsanity.com/paper/1810.13066/) (2018)
- [High-dimensional learning of linear causal networks via inverse covariance estimation](https://mlsanity.com/paper/1311.3492/) (2013)
- [BCD Nets: Scalable Variational Approaches for Bayesian Causal Discovery](https://mlsanity.com/paper/2112.02761/) (2021)
- [DAGs with No Fears: A Closer Look at Continuous Optimization for Learning Bayesian Networks](https://mlsanity.com/paper/2010.09133/) (2020)
- [Differentiable DAG Sampling](https://mlsanity.com/paper/2203.08509/) (2022)
- [Towards Scalable Bayesian Learning of Causal DAGs](https://mlsanity.com/paper/2010.00684/) (2020)
- [Optimizing NOTEARS Objectives via Topological Swaps](https://mlsanity.com/paper/2305.17277/) (2023)
- [CoLiDE: Concomitant Linear DAG Estimation](https://mlsanity.com/paper/2310.02895/) (2023)
- [Inexact Block Coordinate Descent Algorithms for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/1905.04211/) (2019)
- [Causal Fourier Analysis on Directed Acyclic Graphs and Posets](https://mlsanity.com/paper/2209.07970/) (2022)
- [Global Optimality in Bivariate Gradient-based DAG Learning](https://mlsanity.com/paper/2306.17378/) (2023)
- [DAG Learning on the Permutahedron](https://mlsanity.com/paper/2301.11898/) (2023)
- [Efficient Smoothed Concomitant Lasso Estimation for High Dimensional Regression](https://mlsanity.com/paper/1606.02702/) (2016)
- [Generalized Concomitant Multi-Task Lasso for sparse multimodal regression](https://mlsanity.com/paper/1705.09778/) (2017)
- [Exploiting Non-Negativity in DAG Structure Learning](https://mlsanity.com/paper/2605.19947/) (2026)

## Built on by (most cited first, 2 of 2)

- [From topology learning to graph generation: A unifying perspective](https://mlsanity.com/paper/2609.02286/) (2026)
- [Nonnegative DAG Learning via Concomitant Estimation](https://mlsanity.com/paper/2609.34595/) (2026)

## Abstract (arXiv)

Directed acyclic graphs (DAGs) constitute a central modeling tool to enable principled reasoning about cause-effect interactions in complex systems. However, since the causal structure underlying a group of variables is often unknown and interventions may be infeasible or ethically challenging to implement, there is a need to address the task of inferring DAGs from observational data. However, most classical structure identification approaches face two key obstacles: the combinatorial challenge of enforcing acyclicity, which severely limits scalability, and identifiability challenges arising from latent confounding or heterogeneous noise. This tutorial offers an overview of recent signal processing and optimization advances that address these issues by recasting DAG structure learning as a continuous, score-based estimation problem over adjacency matrices. We begin with a didactic introduction to structural equation models and the formulation of causal graph recovery, followed by a historical survey of score-based methods ranging from early combinatorial search schemes and greedy heuristics to modern continuous frameworks that leverage smooth characterizations of acyclicity. Building on this foundation, we describe concomitant DAG estimation methods that jointly infer sparse causal structure and exogenous noise levels, improving robustness under heteroscedasticity and distribution shifts by rendering the estimator noise adaptive. All in all, the tutorial introduces readers to challenges and opportunities for signal processing research at the crossroads of causal inference, high-dimensional statistics, and scalable graph learning, while outlining emerging directions including online, nonlinear, and neural causal discovery.