We prove nearly matching upper and lower bounds on the randomized communication complexity of the following problem: Alice and Bob are each given a probability distribution over $n$ elements, and they wish to estimate within $\pm\epsilon$ the statistical (total variation) distance between their distributions. For some range of parameters, there ... more >>>

Let $D$ be a $b$-wise independent distribution over
$\{0,1\}^m$. Let $E$ be the ``noise'' distribution over
$\{0,1\}^m$ where the bits are independent and each bit is 1
with probability $\eta/2$. We study which tests $f \colon
\{0,1\}^m \to [-1,1]$ are $\e$-fooled by $D+E$, i.e.,
$|\E[f(D+E)] - \E[f(U)]| \le \e$ where ... more >>>

We present a new methodology for proving distribution testing lower bounds, establishing a connection between distribution testing and the simultaneous message passing (SMP) communication model. Extending the framework of Blais, Brody, and Matulef [BBM12], we show a simple way to reduce (private-coin) SMP problems to distribution testing problems. This method ... more >>>