Efficiently Learning Significant Fourier Feature Pairs for Statistical Independence Testing

We propose a novel method to efficiently learn significant Fourier feature pairs for maximizing the power of Hilbert-Schmidt Independence Criterion~(HSIC) based independence tests. We first reinterpret HSIC in the frequency domain, which reveals its limited discriminative power due to the inability to adapt to specific frequency-domain features under the current inflexible configuration. To remedy this shortcoming, we introduce a module of learnable Fourier features, thereby developing a new criterion. We then derive a finite sample estimate of the test power by modeling the behavior of the criterion, thus formulating an optimization objective for significant Fourier feature pairs learning. We show that this optimization objective can be computed in linear time (with respect to the sample size $n$), which ensures fast independence tests. We also prove the convergence property of the optimization objective and establish the consistency of the independence tests. Extensive empirical evaluation on both synthetic and real datasets validates our method's superiority in effectiveness and efficiency, particularly in handling high-dimensional data and dealing with large-scale scenarios.