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

arXiv:2605.05743v2 Announce Type: replace Abstract: 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 $O(nm^2 + m^3)$ 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.