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

Statistics > Machine Learning Title:Fourier Feature Methods for Nonlinear Causal Discovery: FFML Scoring and FFCI Testing in Mixed Data View PDF HTML (experimental)Abstract:Gaussian process marginal likelihood scores and kernel conditional independence tests are theoretically appealing for nonlinear causal discovery but computationally prohibitive at scale. We present two 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 + discrete) parent sets via a product-kernel construction, with a Kronecker path for small discrete parent sets and a Hadamard-product path otherwise. The Fourier Feature Conditional Independence (FFCI) test is a fast nonparametric CI test for mixed data. Each variable is featurized individually: continuous variables via RFF or Orthogonal Random Features (ORF), discrete variables via a Cholesky-factored categorical feature map, with blocks concatenated. Conditioning uses ridge residualization in feature space; the test statistic is a Frobenius norm of the residualized cross-covariance, approximated as a weighted sum of chi-squared variables. Although FFML and FFCI share the same RFF/ORF machinery, they differ architecturally: FFML builds a joint kernel over a parent set for scoring, while FFCI featurizes variables individually for testing. We compare FFML to TRFF, a penalized Student-t regression alternative. Empirically, BOSS+FFML outperforms linear and kernel-ridge baselines on nonlinear data. When run through the same PC-Max implementation, FFCI and RCIT exhibit complementary precision-recall profiles: RCIT is more precise while FFCI achieves better recall and lower SHD, and runs in one third the time. Current browse context: Bibliographic and Citation Tools Code, Data and Media Associated with this Article Demos Recommenders and Search Tools arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.