High-Dimensional Independence Testing and Maximum Marginal Correlation

The statistical hypothesis for testing independence is formulated as: H 0: F XY F XF Y, H A: F XY null F XF Y. T raditional correlation measures like Pearson's correlation [13] are widely used but not applicable to detect nonlinear and high-dimensional dependence structures, whereas many recently proposed dependence measures are able to discover any dependence structure given sufficiently sample size. The most prominent pioneers are the distance correlation [22, 25] and the Hilbert-Schmidt independence criterion [4, 5]. They are shown to be asymptotically 0 if and only if independence, share similar formulation and properties [16, 19], and is valid and consistent for testing independence against any joint distribution at any fixed dimensionality . Other dependence measures are later proposed to improve the finite-sample testing power against strong nonlinear dependencies, such as the Heller-Heller-Gorfine method [6, 7], the multiscale graph correlation [18, 26], among others. A dependence measure can be useful in plenty statistical tasks, including two-sample testing [17], feature screening [10, 27, 30], time-series [2, 12, 31], conditional independence [3, 24, 28], clustering [15, 21], graph testing [9, 29], etc.