# Scalable Causal Discovery from Recursive Nonlinear Data via Truncated Basis Function Scores and Tests

arXiv: https://arxiv.org/abs/2510.04276 · HTML: https://mlsanity.com/paper/2510.04276/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 32 references that are in the map.
- First posted on arXiv 5 Oct 2025.

## Builds on (most cited first, 11 of 11)

- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Causal Effect Inference with Deep Latent-Variable Models](https://mlsanity.com/paper/1705.08821/) (2017)
- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [CAM: Causal additive models, high-dimensional order search and penalized regression](https://mlsanity.com/paper/1310.1533/) (2013)
- [Causal-learn: Causal Discovery in Python](https://mlsanity.com/paper/2307.16405/) (2023)
- [Approximate Kernel-based Conditional Independence Tests for Fast Non-Parametric Causal Discovery](https://mlsanity.com/paper/1702.03877/) (2017)
- [Greedy Relaxations of the Sparsest Permutation Algorithm](https://mlsanity.com/paper/2206.05421/) (2022)
- [Neural Network with Unbounded Activation Functions is Universal Approximator](https://mlsanity.com/paper/1505.03654/) (2015)
- [Fast Scalable and Accurate Discovery of DAGs Using the Best Order Score Search and Grow-Shrink Trees](https://mlsanity.com/paper/2310.17679/) (2023)
- [Improving Accuracy and Scalability of the PC Algorithm by Maximizing P-value](https://mlsanity.com/paper/1610.00378/) (2016)
- [The causal structure of galactic astrophysics](https://mlsanity.com/paper/2510.01112/) (2025)

## Built on by (most cited first, 1 of 1)

- [EML-CD: Causal Mechanism Recovery via EML Symbolic Trees in Structure Learning](https://mlsanity.com/paper/2606.05942/) (2026)

## Abstract (arXiv)

Learning graphical conditional independence structures from nonlinear, continuous or mixed data is a central challenge in machine learning and the sciences, and many existing methods struggle to scale to thousands of samples or hundreds of variables. We introduce two basis-expansion tools for scalable causal discovery. First, the Basis Function BIC (BF-BIC) score uses truncated additive expansions to approximate nonlinear dependencies. BF-BIC is theoretically consistent under additive models and extends to post-nonlinear (PNL) models via an invertible reparameterization. It remains robust under moderate interactions and supports mixed data through a degenerate-Gaussian embedding for discrete variables. In simulations with fully nonlinear neural causal models (NCMs), BF-BIC outperforms kernel- and constraint-based methods (e.g., KCI, RFCI) in both accuracy and runtime. Second, the Basis Function Likelihood Ratio Test (BF-LRT) provides an approximate conditional independence test that is substantially faster than kernel tests while retaining competitive accuracy. Extensive simulations and a real-data application to Canadian wildfire risk show that, when integrated into hybrid searches, BF-based methods enable interpretable and scalable causal discovery. Implementations are available in Python, R, and Java.