# BUILD with Precision: Bottom-Up Inference of Linear DAGs

arXiv: https://arxiv.org/abs/2512.16111 · HTML: https://mlsanity.com/paper/2512.16111/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 29 references that are in the map.
- First posted on arXiv 18 Dec 2025.

## Builds on (most cited first, 18 of 18)

- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [DAG-GNN: DAG Structure Learning with Graph Neural Networks](https://mlsanity.com/paper/1904.10098/) (2019)
- [On the Role of Sparsity and DAG Constraints for Learning Linear DAGs](https://mlsanity.com/paper/2006.10201/) (2020)
- [CAM: Causal additive models, high-dimensional order search and penalized regression](https://mlsanity.com/paper/1310.1533/) (2013)
- [Graph Learning: A Survey](https://mlsanity.com/paper/2105.00696/) (2021)
- [DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization](https://mlsanity.com/paper/2209.08037/) (2022)
- [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)
- [On Causal Discovery with Equal Variance Assumption](https://mlsanity.com/paper/1807.03419/) (2018)
- [A polynomial-time algorithm for learning nonparametric causal graphs](https://mlsanity.com/paper/2006.11970/) (2020)
- [Learning Identifiable Gaussian Bayesian Networks in Polynomial Time and Sample Complexity](https://mlsanity.com/paper/1703.01196/) (2017)
- [Learning Some Popular Gaussian Graphical Models without Condition Number Bounds](https://mlsanity.com/paper/1905.01282/) (2019)
- [Optimal estimation of Gaussian DAG models](https://mlsanity.com/paper/2201.10548/) (2022)
- [CoLiDE: Concomitant Linear DAG Estimation](https://mlsanity.com/paper/2310.02895/) (2023)
- [Information-theoretic limits of Bayesian network structure learning](https://mlsanity.com/paper/1601.07460/) (2016)
- [Non-negative Weighted DAG Structure Learning](https://mlsanity.com/paper/2409.07880/) (2024)
- [Learning Gaussian DAG Models without Condition Number Bounds](https://mlsanity.com/paper/2511.06164/) (2025)

## Built on by (most cited first, 4 of 4)

- [Exploiting Non-Negativity in DAG Structure Learning](https://mlsanity.com/paper/2605.19947/) (2026)
- [Causal Discovery in Equal Variance Linear Gaussian DAGs via SURE-Tuned Ridge Regression](https://mlsanity.com/paper/2608.17132/) (2026)
- [Statistical Testing on Directed Graphs by Surrogate Data Generation](https://mlsanity.com/paper/2606.00758/) (2026)
- [Nonnegative DAG Learning via Concomitant Estimation](https://mlsanity.com/paper/2609.34595/) (2026)

## Abstract (arXiv)

Learning the structure of directed acyclic graphs (DAGs) from observational data is a central problem in causal discovery, statistical signal processing, and machine learning. Under a linear Gaussian structural equation model (SEM) with equal noise variances, the problem is identifiable and we show that the ensemble precision matrix of the observations exhibits a distinctive structure that facilitates DAG recovery. Exploiting this property, we propose BUILD (Bottom-Up Inference of Linear DAGs), a deterministic stepwise algorithm that identifies leaf nodes and their parents, then prunes the leaves by removing incident edges to proceed to the next step, exactly reconstructing the DAG from the true precision matrix. In practice, precision matrices must be estimated from finite data, and ill-conditioning may lead to error accumulation across BUILD steps. As a mitigation strategy, we periodically re-estimate the precision matrix (with less variables as leaves are pruned), trading off runtime for enhanced robustness. Reproducible results on challenging synthetic benchmarks demonstrate that BUILD compares favorably to state-of-the-art DAG learning algorithms, while offering an explicit handle on complexity.