Enhanced HSIC for independence test via projection integration

Abstract

Among the various measures of dependence between two random vectors, the Hilbert–Schmidt independence criterion (HSIC) is widely recognized and has gained significant attention in recent years. However, HSIC-based tests can become less effective as dimensionality increases and nonlinear dependencies become more complex. In this paper, we introduce a novel method that integrates the HSIC with a Gaussian kernel over all one-dimensional projections. The resulting metric has a closed-form expression, is non-negative, and equals zero if and only if the random vectors are independent. We estimate the integrated HSIC using $U$-statistic theory and analyze its asymptotic properties under the null hypothesis and two types of alternative hypotheses. Comprehensive numerical studies demonstrate that our method preserves the advantages of HSIC in univariate settings while effectively capturing complex nonlinear dependencies as dimensionality increases.