Locally Sampleable Uniform Symmetric Distributions
Daniel M. Kane, Anthony Ostuni, Kewen Wu
TL;DR
The paper characterizes the distributions that constant-depth Boolean circuits can sample when given uniform inputs, proving that any $d$-local sampler whose output is close to a uniform symmetric distribution must be close to one of six canonical distributions (zeros, ones, zerones, evens, odds, all). The approach combines a hypergraph-based structured decomposition with a local limit theorem, distinguishing tail and central regimes to reduce the problem to analyzing parity and weight distributions. A Kolmogorov-type decomposition and a continuity bound underpin a local limit theorem that shows near-centered targets are well-approximated by even/odd mixtures, enabling a complete, depth-independent classification. These results tightens our understanding of the sampling power of $\mathsf{NC^0}$ circuits and has implications for pseudorandomness, data structures, and sampling in the low-depth regime.
Abstract
We characterize the power of constant-depth Boolean circuits in generating uniform symmetric distributions. Let $f\colon\{0,1\}^m\to\{0,1\}^n$ be a Boolean function where each output bit of $f$ depends only on $O(1)$ input bits. Assume the output distribution of $f$ on uniform input bits is close to a uniform distribution $D$ with a symmetric support. We show that $D$ is essentially one of the following six possibilities: (1) point distribution on $0^n$, (2) point distribution on $1^n$, (3) uniform over $\{0^n,1^n\}$, (4) uniform over strings with even Hamming weights, (5) uniform over strings with odd Hamming weights, and (6) uniform over all strings. This confirms a conjecture of Filmus, Leigh, Riazanov, and Sokolov (RANDOM 2023).