Testing Closeness of Multivariate Distributions via Ramsey Theory

We investigate the statistical task of closeness (or equivalence) testing for multidimensional distributions. Specifically, given sample access to two unknown distributions $\mathbf p, \mathbf q$ on $\mathbb R^d$, we want to distinguish between the case that $\mathbf p=\mathbf q$ versus $|\mathbf p-\mathbf q|{Ak} > \epsilon$, where $|\mathbf p-\mathbf q|{Ak}$ denotes the generalized ${A}k$ distance between $\mathbf p$ and $\mathbf q$ -- measuring the maximum discrepancy between the distributions over any collection of $k$ disjoint, axis-aligned rectangles. Our main result is the first closeness tester for this problem with {\em sub-learning} sample complexity in any fixed dimension and a nearly-matching sample complexity lower bound. In more detail, we provide a computationally efficient closeness tester with sample complexity $O\left((k^{6/7}/ \mathrm{poly}d(\epsilon)) \log^{d(k)\right)$.} On the lower bound side, we establish a qualitatively matching sample complexity lower bound of $\Omega(k^{{6/7}/\mathrm{poly}(\epsilon))$,} even for $d=2$. These sample complexity bounds are surprising because the sample complexity of the problem in the univariate setting is $\Theta(k^{{4/5}/\mathrm{poly}(\epsilon))$.} This has the interesting consequence that the jump from one to two dimensions leads to a substantial increase in sample complexity, while increases beyond that do not. As a corollary of our general $Ak$ tester, we obtain $d{\mathrm TV}$-closeness testers for pairs of $k$-histograms on $\mathbb R^d$ over a common unknown partition, and pairs of uniform distributions supported on the union of $k$ unknown disjoint axis-aligned rectangles. Both our algorithm and our lower bound make essential use of tools from Ramsey theory.