# Concave Penalized Estimation of Sparse Gaussian Bayesian Networks

arXiv: https://arxiv.org/abs/1401.0852 · HTML: https://mlsanity.com/paper/1401.0852/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jan 2014.

## Built on by (most cited first, 28 of 28)

- [D'ya like DAGs? A Survey on Structure Learning and Causal Discovery](https://mlsanity.com/paper/2103.02582/) (2021)
- [Beware of the Simulated DAG! Causal Discovery Benchmarks May Be Easy To Game](https://mlsanity.com/paper/2102.13647/) (2021)
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- [Optimizing NOTEARS Objectives via Topological Swaps](https://mlsanity.com/paper/2305.17277/) (2023)
- [Simultaneous Missing Value Imputation and Structure Learning with Groups](https://mlsanity.com/paper/2110.08223/) (2021)
- [Distributed Learning of Generalized Linear Causal Networks](https://mlsanity.com/paper/2201.09194/) (2022)
- [Causal Fourier Analysis on Directed Acyclic Graphs and Posets](https://mlsanity.com/paper/2209.07970/) (2022)
- [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)
- [Markov Equivalence and Consistency in Differentiable Structure Learning](https://mlsanity.com/paper/2410.06163/) (2024)
- [iSCAN: Identifying Causal Mechanism Shifts among Nonlinear Additive Noise Models](https://mlsanity.com/paper/2306.17361/) (2023)
- [Sequentially learning the topological ordering of causal directed acyclic graphs with likelihood ratio scores](https://mlsanity.com/paper/2202.01748/) (2022)
- [STICC: A multivariate spatial clustering method for repeated geographic pattern discovery with consideration of spatial contiguity](https://mlsanity.com/paper/2203.09611/) (2022)
- [Do learned representations respect causal relationships?](https://mlsanity.com/paper/2204.00762/) (2022)
- [Learning Sparse Fixed-Structure Gaussian Bayesian Networks](https://mlsanity.com/paper/2107.10450/) (2021)
- [Learning and interpreting asymmetry-labeled DAGs: a case study on COVID-19 fear](https://mlsanity.com/paper/2301.00629/) (2023)
- [Learning complex dependency structure of gene regulatory networks from high dimensional micro-array data with Gaussian Bayesian networks](https://mlsanity.com/paper/2106.15365/) (2021)
- [Learning Bayesian Networks through Birkhoff Polytope: A Relaxation Method](https://mlsanity.com/paper/2107.01658/) (2021)
- [A Stochastic Variance-Reduced Coordinate Descent Algorithm for Learning Sparse Bayesian Network from Discrete High-Dimensional Data](https://mlsanity.com/paper/2108.09501/) (2021)
- [Identifying Causal Influences on Publication Trends and Behavior: A Case Study of the Computational Linguistics Community](https://mlsanity.com/paper/2110.07938/) (2021)
- [Recovering Linear Causal Models with Latent Variables via Cholesky Factorization of Covariance Matrix](https://mlsanity.com/paper/2311.00674/) (2023)
- [An Asymptotically Optimal Coordinate Descent Algorithm for Learning Bayesian Networks from Gaussian Models](https://mlsanity.com/paper/2408.11977/) (2024)
- [Inconsistency of cross-validation for structure learning in Gaussian graphical models](https://mlsanity.com/paper/2312.17047/) (2023)
- [Graph Structure Inference with BAM: Introducing the Bilinear Attention Mechanism](https://mlsanity.com/paper/2402.07735/) (2024)
- [Causal Discovery on Dependent Binary Data](https://mlsanity.com/paper/2412.20289/) (2024)
- [Scalable Structure Learning of Bayesian Networks by Learning Algorithm Ensembles](https://mlsanity.com/paper/2506.22848/) (2025)
- [DAG-CLIP: A DAG Learning Framework in the Presence of Latent Variables](https://mlsanity.com/paper/2610.06086/) (2026)

## Abstract (arXiv)

We develop a penalized likelihood estimation framework to estimate the structure of Gaussian Bayesian networks from observational data. In contrast to recent methods which accelerate the learning problem by restricting the search space, our main contribution is a fast algorithm for score-based structure learning which does not restrict the search space in any way and works on high-dimensional datasets with thousands of variables. Our use of concave regularization, as opposed to the more popular ℓ0 (e.g. BIC) penalty, is new. Moreover, we provide theoretical guarantees which generalize existing asymptotic results when the underlying distribution is Gaussian. Most notably, our framework does not require the existence of a so-called faithful DAG representation, and as a result the theory must handle the inherent nonidentifiability of the estimation problem in a novel way. Finally, as a matter of independent interest, we provide a comprehensive comparison of our approach to several standard structure learning methods using open-source packages developed for the R language. Based on these experiments, we show that our algorithm is significantly faster than other competing methods while obtaining higher sensitivity with comparable false discovery rates for high-dimensional data. In particular, the total runtime for our method to generate a solution path of 20 estimates for DAGs with 8000 nodes is around one hour.