We characterize the symmetric distributions that can be (approximately) generated by shallow Boolean circuits. More precisely, let $f\colon \{0,1\}^m \to \{0,1\}^n$ be a Boolean function where each output bit depends on at most $d$ input bits. Suppose the output distribution of $f$ evaluated on uniformly random input bits is close ... more >>>

The original proof of the PCP Theorem composes a Reed-Muller-based PCP with itself, and then composes the resulting PCP with a Hadamard-based PCP [Arora, Lund, Motwani, Sudan and Szegedy ({\em JACM}, 1998)].
Hence, that proof applies a (general) proof composition result twice.
(Dinur's alternative proof consists of logarithmically many gap ... more >>>

One of the most fundamental problems in the field of hypothesis testing is the identity testing problem: whether samples from some unknown distribution $\mathcal{G}$ are actually from some explicit distribution $\mathcal{D}$. It is known that when the distribution $\mathcal{D}$ has support $[N]$, the optimal sample complexity for the identity testing ... more >>>