Symmetric Distributions from Shallow Circuits

TL;DR

This paper characterizes the symmetric distributions that shallow Boolean circuits can generate, showing they are close to mixtures of simple distributions like uniform distributions over even or odd Hamming weight strings and biased product distributions, with structure determined by low-degree polynomials.

Contribution

It extends the classification of distributions generated by shallow circuits to include non-uniform symmetric distributions, revealing their structure as mixtures of basic distributions with polynomial-based weights.

Findings

Symmetric distributions generated by shallow circuits are close to mixtures of specific simple distributions.

The mixture weights are determined by low-degree, sparse polynomials over GF(2).

This generalizes previous results limited to uniform symmetric distributions.

Abstract

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 in total variation distance to a symmetric distribution $\mathcal{D}$ over $\{0,1\}^n$. Then $\mathcal{D}$ must be close to a mixture of the uniform distribution over $n$-bit strings of even Hamming weight, the uniform distribution over $n$-bit strings of odd Hamming weight, and $\gamma$-biased product distributions for $\gamma$ an integer multiple of $2^{-d}$. Moreover, the mixing weights are determined by low-degree, sparse $\mathbb{F}_2$-polynomials. This extends the previous classification for generating symmetric distributions that are also uniform over their support.