Measuring Statistical Dependence via Characteristic Function IPM

We study statistical dependence in the frequency domain using the integral probability metric (IPM) framework. We propose the uniform Fourier dependence measure (UFDM) defined as the uniform norm of the difference between the joint and product-marginal characteristic functions. We provide a theoretical analysis, highlighting key properties, such as invariances, monotonicity in linear dimension reduction, and a concentration bound. For the estimation of the UFDM, we propose a gradient-based algorithm with singular value decomposition (SVD) warm-up and show that this warm-up is essential for stable performance. The empirical estimator of UFDM is differentiable, and it can be integrated into modern machine learning pipelines. In experiments with synthetic and real-world data, we compare UFDM with distance correlation (DCOR), Hilbert-Schmidt independence criterion (HSIC), and matrix-based Rényi's α-entropy functional (MEF) in permutation-based statistical independence testing and supervised feature extraction. Independence test experiments showed the effectiveness of UFDM at detecting some sparse geometric dependencies in a diverse set of patterns that span different linear and nonlinear interactions, including copulas and geometric structures. In feature extraction experiments across 16 OpenML datasets, we conducted 160 pairwise comparisons: UFDM statistically significantly outperformed other baselines in 20 cases and was outperformed in 13.