Testing Closeness of Multivariate Distributions via Ramsey Theory
This addresses a fundamental statistical testing problem for high-dimensional data, with implications for machine learning and data analysis, though it is incremental in extending univariate methods to multivariate settings.
The paper tackles the problem of closeness testing for multivariate distributions using the generalized A_k distance, achieving the first sub-learning sample complexity in fixed dimensions with an efficient tester at O((k^{6/7}/poly_d(ε)) log^d(k)) and a matching lower bound of Ω(k^{6/7}/poly(ε)), showing a substantial increase from univariate to two dimensions.
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 $\|\mathbf p-\mathbf q\|_{A_k} > ε$, where $\|\mathbf p-\mathbf q\|_{A_k}$ 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 $O\left((k^{6/7}/ \mathrm{poly}_d(ε)) \log^d(k)\right)$. On the lower bound side, we establish a qualitatively matching sample complexity lower bound of $Ω(k^{6/7}/\mathrm{poly}(ε))$, even for $d=2$. These sample complexity bounds are surprising because the sample complexity of the problem in the univariate setting is $Θ(k^{4/5}/\mathrm{poly}(ε))$. 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 $A_k$ 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.