Computationally efficient permutation tests for the multivariate two-sample problem based on energy distance or maximum mean discrepancy statistics

Non-parametric two-sample tests based on energy distance or maximum mean discrepancy are widely used statistical tests for comparing multivariate data from two populations. While these tests enjoy desirable statistical properties, their test statistics can be expensive to compute as they require the computation of 3 distinct Euclidean distance (or kernel) matrices between samples, where the time complexity of each of these computations (namely, $O(n_{x}^2 p)$, $O(n_{y}^2 p)$, and $O(n_{x} n_{y} p)$) scales quadratically with the number of samples ($n_x$, $n_y$) and linearly with the number of variables ($p$). Since the standard permutation test requires repeated re-computations of these expensive statistics it's application to large datasets can become unfeasible. While several statistical approaches have been proposed to mitigate this issue, they all sacrifice desirable statistical properties to decrease the computational cost (e.g., trade computation speed by a decrease in statistical power). A better computational strategy is to first pre-compute the Euclidean distance (kernel) matrix of the concatenated data, and then permute indexes and retrieve the corresponding elements to compute the re-sampled statistics. While this strategy can reduce the computation cost relative to the standard permutation test, it relies on the computation of a larger Euclidean distance (kernel) matrix with complexity $O((n_x + n_y)^2 p)$. In this paper, we present a novel computationally efficient permutation algorithm which only requires the pre-computation of the 3 smaller matrices and achieves large computational speedups without sacrificing finite-sample validity or statistical power. We illustrate its computational gains in a series of experiments and compare its statistical power to the current state-of-the-art approach for balancing computational cost and statistical performance.