Identity Testing for High-Dimensional Distributions via Entropy Tensorization

We present improved algorithms and matching statistical and computational lower bounds for the problem of identity testing $n$-dimensional distributions. In the identity testing problem, we are given as input an explicit distribution $\mu$, an $\varepsilon>0$, and access to a sampling oracle for a hidden distribution $\pi$. The goal 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 may be needed, and hence previous works have studied identity testing with additional access to various conditional sampling oracles. We consider here a significantly weaker conditional sampling oracle, called the Coordinate Oracle, and provide a fairly complete 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 the visible distribution $\mu$, then there is an efficient identity testing algorithm for any hidden $\pi$ that uses $\tilde{O}(n/\varepsilon)$ queries to the Coordinate Oracle. Approximate tensorization of entropy is a classical tool for proving optimal mixing time bounds of Markov chains for high-dimensional distributions, and recently has been established for many families of distributions via spectral independence. We complement our algorithmic result for identity testing with a matching $\Omega(n/\varepsilon)$ statistical lower bound for the number of queries under the Coordinate Oracle. We also prove a computational phase transition: for sparse antiferromagnetic Ising models over $\{+1,-1\}^n$, in the regime where approximate tensorization of entropy fails, there is no efficient identity testing algorithm unless RP=NP.