  We discuss how MultiFIT, the Multiscale Fisher's Independence Test for
Multivariate Dependence proposed by Gorsky and Ma (2022), compares to existing
linear-time kernel tests based on the Hilbert-Schmidt independence criterion
(HSIC). We highlight the fact that the levels of the kernel tests at any finite
sample size can be controlled exactly, as it is the case with the level of
MultiFIT. In our experiments, we observe some of the performance limitations of
MultiFIT in terms of test power.
