🤖 AI Summary
To address the computational bottleneck of two-sample testing in high-dimensional non-Euclidean spaces, this paper proposes an efficient Maximum Mean Discrepancy (MMD) test based on Random Fourier Features (RFF) and spectral regularization. The method reduces time complexity from $O(n^3)$ to $O(n^2)$. It provides the first rigorous proof that RFF-based MMD attains the minimax optimal statistical rate. We further introduce a data-adaptive strategy for selecting the spectral regularization parameter and optimizing the kernel, and construct a practical permutation testing framework. Theoretical analysis, grounded in eigenvalue decay properties of integral operators, characterizes the bias–variance trade-off. Experiments on synthetic and multiple benchmark datasets demonstrate substantial computational speedup while maintaining high statistical power—achieving only a marginal loss (<3%) in test efficacy compared to the exact MMD test, and outperforming existing accelerated kernel two-sample tests.
📝 Abstract
Reproducing Kernel Hilbert Space (RKHS) embedding of probability distributions has proved to be an effective approach, via MMD (maximum mean discrepancy) for nonparametric hypothesis testing problems involving distributions defined over general (non-Euclidean) domains. While a substantial amount of work has been done on this topic, only recently, minimax optimal two-sample tests have been constructed that incorporate, unlike MMD, both the mean element and a regularized version of the covariance operator. However, as with most kernel algorithms, the computational complexity of the optimal test scales cubically in the sample size, limiting its applicability. In this paper, we propose a spectral regularized two-sample test based on random Fourier feature (RFF) approximation and investigate the trade-offs between statistical optimality and computational efficiency. We show the proposed test to be minimax optimal if the approximation order of RFF (which depends on the smoothness of the likelihood ratio and the decay rate of the eigenvalues of the integral operator) is sufficiently large. We develop a practically implementable permutation-based version of the proposed test with a data-adaptive strategy for selecting the regularization parameter and the kernel. Finally, through numerical experiments on simulated and benchmark datasets, we demonstrate that the proposed RFF-based test is computationally efficient and performs almost similar (with a small drop in power) to the exact test.