  We study the tolerant testing problem for high-dimensional samplers. Given as
input two samplers $\mathcal{P}$ and $\mathcal{Q}$ over the $n$-dimensional
space $\{0,1\}^n$, and two parameters $\varepsilon_2 > \varepsilon_1$, the goal
of tolerant testing is to test whether the distributions generated by
$\mathcal{P}$ and $\mathcal{Q}$ are $\varepsilon_1$-close or
$\varepsilon_2$-far. Since exponential lower bounds (in $n$) are known for the
problem in the standard sampling model, research has focused on models where
one can draw \textit{conditional} samples.
  Among these models, \textit{subcube conditioning} ($\mathsf{SUBCOND}$), which
allows conditioning on arbitrary subcubes of the domain, holds the promise of
widespread adoption in practice owing to its ability to capture the natural
behavior of samplers in constrained domains. To translate the promise into
practice, we need to overcome two crucial roadblocks for tests based on
$\mathsf{SUBCOND}$: the prohibitively large number of queries
($\tilde{\mathcal{O}}(n^5/\varepsilon_2^5)$) and limitation to non-tolerant
testing (i.e., $\varepsilon_1 = 0$).
  The primary contribution of this work is to overcome the above challenges: we
design a new tolerant testing methodology (i.e., $\varepsilon_1 \geq 0$) that
allows us to significantly improve the upper bound to
$\tilde{\mathcal{O}}(n^3/(\varepsilon_2-\varepsilon_1)^5)$.
