Do You Want to Know if Two Distributions Are Close to Each Other?Testing the Closeness With Statistical Significance

Are two distributions close to each other with statistical significance? Distribution closeness testing (DCT) formalizes this question by testing whether the distance between a distribution pair is at least $\epsilon$-far. Existing DCT methods mainly measure discrepancies between a distribution pair defined on discrete spaces (e.g., using total variation), which limits their applications to complex data (e.g., images). To extend DCT to more types of data, a natural idea is to introduce maximum mean discrepancy (MMD), a powerful measurement of the distributional discrepancy between two complex distributions, into DCT scenarios. However, we find that MMD's value can be the same for many pairs of distributions that have different norms in the same reproducing kernel Hilbert space (RKHS), which potentially have different closeness levels, making MMD less informative when assessing the closeness of multiple distribution pairs. To mitigate the issue, we design a new measurement of distributional discrepancy, norm-adaptive MMD (NAMMD), which scales MMD's value using the RKHS norms of distributions. Based on the asymptotic distribution of NAMMD, we finally propose the NAMMD-based DCT to assess the closeness level of a distribution pair. Theoretically, we prove that NAMMD-based DCT has higher test power compared to MMD-based DCT, with bounded type-I error, which is also validated by extensive experiments on many types of data (e.g., synthetic noise, real images).