# Fourier Feature Methods for Nonlinear Causal Discovery: FFML Scoring, TRFF Scoring, and FFCI Testing in Mixed Data

arXiv: https://arxiv.org/abs/2605.05743 · HTML: https://mlsanity.com/paper/2605.05743/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 13 references that are in the map.
- First posted on arXiv 7 May 2026.

## Builds on (most cited first, 7 of 7)

- [Causal Inference and Causal Explanation with Background Knowledge](https://mlsanity.com/paper/1302.4972/) (2013)
- [Causal-learn: Causal Discovery in Python](https://mlsanity.com/paper/2307.16405/) (2023)
- [Orthogonal Random Features](https://mlsanity.com/paper/1610.09072/) (2016)
- [Approximate Kernel-based Conditional Independence Tests for Fast Non-Parametric Causal Discovery](https://mlsanity.com/paper/1702.03877/) (2017)
- [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)
- [Py-Tetrad and RPy-Tetrad: A New Python Interface with R Support for Tetrad Causal Search](https://mlsanity.com/paper/2308.07346/) (2023)

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

- [GFCM: A Tail-Sensitive Mixed-Type Conditional Independence Test for Causal Discovery](https://mlsanity.com/paper/2608.15332/) (2026)

## Abstract (arXiv)

Gaussian process (GP) marginal likelihood scores and kernel conditional independence tests are theoretically appealing for nonlinear causal discovery but computationally prohibitive at scale. We present three complementary RFF-based methods forming a practical toolkit for score-based, constraint-based, and hybrid causal discovery. The Fourier Feature Marginal Likelihood (FFML) score approximates the exact GP marginal likelihood by replacing the n x n kernel Gram matrix with a finite-dimensional feature representation, reducing cost to O(nm2 + m3) while retaining the probabilistic interpretation and automatic complexity penalty of the exact score. FFML extends to mixed (continuous and discrete) parent sets via a product-kernel construction, with a Kronecker path for small discrete parent sets and a Hadamard-product path otherwise. The Tetrad Random Fourier Feature (TRFF) score is a complementary BIC-style alternative using penalized Student-t regression with random Fourier features. TRFF offers robustness to heavy-tailed noise and faster runtime than FFML. Empirically, TRFF and FFML exhibit a complementary precision-recall profile: TRFF achieves higher precision while FFML achieves better recall and lower SHD overall. The Fourier Feature Conditional Independence (FFCI) test is a fast nonparametric CI test for mixed data, using ridge residualization in feature space and a Frobenius-norm cross-covariance statistic approximated as a weighted sum of chi-squared variables. Empirically, BOSS+FFML achieves the lowest SHD on nonlinear data, while BOSS+TRFF offers the highest precision. When run through PC-Max, FFCI and RCIT exhibit complementary precision-recall profiles: RCIT is more precise while FFCI achieves better recall and substantially lower SHD, at approximately twice the runtime.