Locally Sampleable Uniform Symmetric Distributions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917936128589824 |
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| author | Kane, Daniel M. Ostuni, Anthony Wu, Kewen |
| author_facet | Kane, Daniel M. Ostuni, Anthony Wu, Kewen |
| contents | 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_08183 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Locally Sampleable Uniform Symmetric Distributions Kane, Daniel M. Ostuni, Anthony Wu, Kewen Computational Complexity 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). |
| title | Locally Sampleable Uniform Symmetric Distributions |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2411.08183 |