用傅里叶特征加速非线性因果发现，三方法兼顾效率与精度。
Fourier Feature Methods for Nonlinear Causal Discovery: FFML Scoring, TRFF Scoring, and FFCI Testing in Mixed Data
- 用随机傅里叶特征近似高斯过程得分，降低计算复杂度至O(nm² + m³)
- TRFF和FFML在精确率与召回率上互补，整体降低结构误差（SHD）
- FFCI测试支持混合数据，比RCIT更快且更低结构误差，适合大规模场景
高斯过程边际似然得分和核条件独立性检验在理论上适合非线性因果发现，但计算成本过高。本文提出三种基于随机傅里叶特征（RFF）的互补方法，构成实用的评分法、约束法和混合因果发现工具包。傅里叶特征边际似然（FFML）通过有限维特征表示替代n×n核格拉姆矩阵，将计算成本降至O(nm² + m³)，同时保留概率解释和自动复杂度惩罚。FFML通过乘积核构造扩展至混合数据（连续与离散变量），小离散父集采用克罗内克路径，其余采用哈达玛积路径。四面体随机傅里叶特征（TRFF）是基于学生分布回归的类BIC替代方案，使用随机傅里叶特征，对重尾噪声更鲁棒，运行速度优于FFML。实验表明，TRFF与FFML在精确率-召回率曲线上互补：TRFF精度更高，而FFML召回率更好，整体SHD更低。傅里叶特征条件独立（FFCI）测试是一种快速非参数条件独立检验方法，适用于混合数据，通过特征空间岭残差化和弗罗贝尼乌斯范数交叉协方差统计量实现，其近似为加权卡方变量之和。实验显示，BOSS+FFML在非线性数据上达到最低SHD，而BOSS+TRFF精度最高。当与PC-Max结合时，FFCI与RCIT表现互补：RCIT更精确，而FFCI召回率更高，结构误差显著更低，运行时间约为两倍。
原文摘要 · Abstract (English)
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.
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