  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.
