We propose novel statistics which maximise the power of a two-sample test
based on the Maximum Mean Discrepancy (MMD), by adapting over the set of
kernels used in defining it. For finite sets, this reduces to combining
(normalised) MMD values under each of these kernels via a weighted soft
maximum. Exponential concentration bounds are proved for our proposed
statistics under the null and alternative. We further show how these kernels
can be chosen in a data-dependent but permutation-independent way, in a
well-calibrated test, avoiding data splitting. This technique applies more
broadly to general permutation-based MMD testing, and includes the use of deep
kernels with features learnt using unsupervised models such as auto-encoders.
We highlight the applicability of our MMD-FUSE test on both synthetic
low-dimensional and real-world high-dimensional data, and compare its
performance in terms of power against current state-of-the-art kernel tests.

本研究提出了一种基于最大均值差异（MMD）的双样本检验，通过适应其核集合，提出了新的统计量。我们进一步展示了这些核如何在无需数据拆分或置换而根据数据来选择，这项技术广泛应用于基于置换的 MMD 检验并包括深度核的使用。我们强调了 MMD-FUSE 测试在合成和实际数据上的适用性，并比较了其在功率方面的表现与当前最先进的核测试。