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Symmetric Distributions from Shallow Circuits

Daniel M. Kane, Anthony Ostuni, Kewen Wu

TL;DR

The paper addresses which symmetric distributions over $\{0,1\\}^n$ can be generated by shallow circuits, focusing on $d$-local Boolean functions. It proves that if a $d$-local function $f$ applied to uniform inputs yields a distribution close to a symmetric target, then the target is well-approximated by a mixture of the uniform distributions over even and odd Hamming weights and $\\gamma$-biased product distributions with $\\gamma$ a dyadic multiple of $2^{-d}$. The mixing weights are shown to be governed by low-degree $\\f{F}_2$-polynomials, and the mixture can be realized by $O_d(1)$-local constructions, generalizing prior uniform-symmetric classifications. The authors develop a four-step framework—removing large influences, Kolmogorov-distance analysis, approximate continuity, and composition—to connect local output structure to global weight distributions and assemble the final characterization. They also discuss learning implications for locally-sampleable symmetric distributions and outline open problems on exact characterization and broader biases, offering a roadmap for exact classification conjectures and potential extensions.

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 $γ$-biased product distributions for $γ$ 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.

Symmetric Distributions from Shallow Circuits

TL;DR

The paper addresses which symmetric distributions over can be generated by shallow circuits, focusing on -local Boolean functions. It proves that if a -local function applied to uniform inputs yields a distribution close to a symmetric target, then the target is well-approximated by a mixture of the uniform distributions over even and odd Hamming weights and -biased product distributions with a dyadic multiple of . The mixing weights are shown to be governed by low-degree -polynomials, and the mixture can be realized by -local constructions, generalizing prior uniform-symmetric classifications. The authors develop a four-step framework—removing large influences, Kolmogorov-distance analysis, approximate continuity, and composition—to connect local output structure to global weight distributions and assemble the final characterization. They also discuss learning implications for locally-sampleable symmetric distributions and outline open problems on exact characterization and broader biases, offering a roadmap for exact classification conjectures and potential extensions.

Abstract

We characterize the symmetric distributions that can be (approximately) generated by shallow Boolean circuits. More precisely, let be a Boolean function where each output bit depends on at most input bits. Suppose the output distribution of evaluated on uniformly random input bits is close in total variation distance to a symmetric distribution over . Then must be close to a mixture of the uniform distribution over -bit strings of even Hamming weight, the uniform distribution over -bit strings of odd Hamming weight, and -biased product distributions for an integer multiple of . Moreover, the mixing weights are determined by low-degree, sparse -polynomials. This extends the previous classification for generating symmetric distributions that are also uniform over their support.

Paper Structure

This paper contains 36 sections, 22 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Our Result
  3. Learning Structured Distributions.
  4. Open Problems
  5. Paper Organization.
  6. Proof Overview
  7. Proof Overview of Theorem \ref{['thm:special_main']}
  8. Step 1: Removing Large Influences.
  9. Step 2: Kolmogorov Distance.
  10. Step 3: Approximate Continuity.
  11. Step 4: Putting It Together.
  12. Mixing Weights.
  13. Technical Comparison to Prior Works
  14. Preliminaries
  15. Asymptotics.
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $\varepsilon\in[0,1]$ be arbitrary. Assume $f\colon\{0,1\}^m\to\{0,1\}^n$ is computable by an $\mathsf{NC^0}$ circuit of constant depth and $f(\mathcal{U}^m)$ is $\varepsilon$-close in total variation distance to a uniform symmetric distribution where $n$ is sufficiently large. Then $f(\mathcal{

Theorems & Definitions (62)

  • Theorem 1.1: kane2024locally2
  • Theorem 1.2: Informal version of \ref{['thm:main']}
  • Example 1.3
  • Conjecture 1.4
  • Lemma 3.2: kane2024locality
  • Lemma 3.3: viola2020sampling
  • Lemma 3.4: kane2024locally2
  • Theorem 4.1
  • Remark 4.2
  • Remark 4.3
  • ...and 52 more