Thinning to improve two-sample discrepancy

The discrepancy between two independent samples $X_1,\dots,X_n$ and $Y_1,\dots,Y_n$ drawn from the same distribution on $\mathbb{R}^d$ typically has order $O(\sqrt{n})$ even in one dimension. We give a simple online algorithm that reduces the discrepancy to $O(\log^{2d} n)$ by discarding a small fraction of the points.