  We study the identity testing problem for high-dimensional distributions.
Given as input an explicit distribution $\mu$, an $\varepsilon>0$, and access
to sampling oracle(s) for a hidden distribution $\pi$, the goal in identity
testing is to distinguish whether the two distributions $\mu$ and $\pi$ are
identical or are at least $\varepsilon$-far apart. When there is only access to
full samples from the hidden distribution $\pi$, it is known that exponentially
many samples (in the dimension) may be needed for identity testing, and hence
previous works have studied identity testing with additional access to various
"conditional" sampling oracles. We consider a significantly weaker conditional
sampling oracle, which we call the $\mathsf{Coordinate\ Oracle}$, and provide a
computational and statistical characterization of the identity testing problem
in this new model.
  We prove that if an analytic property known as approximate tensorization of
entropy holds for an $n$-dimensional visible distribution $\mu$, then there is
an efficient identity testing algorithm for any hidden distribution $\pi$ using
$\tilde{O}(n/\varepsilon)$ queries to the $\mathsf{Coordinate\ Oracle}$.
Approximate tensorization of entropy is a pertinent condition as recent works
have established it for a large class of high-dimensional distributions. We
also prove a computational phase transition: for a well-studied class of
$n$-dimensional distributions, specifically sparse antiferromagnetic Ising
models over $\{+1,-1\}^n$, we show that in the regime where approximate
tensorization of entropy fails, there is no efficient identity testing
algorithm unless $\mathsf{RP}=\mathsf{NP}$. We complement our results with a
matching $\Omega(n/\varepsilon)$ statistical lower bound for the sample
complexity of identity testing in the $\mathsf{Coordinate\ Oracle}$ model.
