  In this work we present novel differentially private identity
(goodness-of-fit) testers for natural and widely studied classes of
multivariate product distributions: Gaussians in $\mathbb{R}^d$ with known
covariance and product distributions over $\{\pm 1\}^{d}$. Our testers have
improved sample complexity compared to those derived from previous techniques,
and are the first testers whose sample complexity matches the order-optimal
minimax sample complexity of $O(d^{1/2}/\alpha^2)$ in many parameter regimes.
We construct two types of testers, exhibiting tradeoffs between sample
complexity and computational complexity. Finally, we provide a two-way
reduction between testing a subclass of multivariate product distributions and
testing univariate distributions, and thereby obtain upper and lower bounds for
testing this subclass of product distributions.
