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Near-optimal lower bounds on the threshold degree and sign-rank of AC⁰

Authors:

Alexander A. Sherstov,

Pei WuAuthors Info & Claims

STOC 2019: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

Pages 401 - 412

https://doi.org/10.1145/3313276.3316408

Published: 23 June 2019 Publication History

Abstract

The threshold degree of a Boolean function f∶{0,1}ⁿ→{0,1} is the minimum degree of a real polynomial p that represents f in sign: sgn p(x)=(−1)^f(x). A related notion is sign-rank, defined for a Boolean matrix F=[F_ij] as the minimum rank of a real matrix M with sgn M_ij=(−1)^F_ij. Determining the maximum threshold degree and sign-rank achievable by constant-depth circuits (AC⁰) is a well-known and extensively studied open problem, with complexity-theoretic and algorithmic applications.

We give an essentially optimal solution to this problem. For any є>0, we construct an AC⁰ circuit in n variables that has threshold degree Ω(n^1−є) and sign-rank exp(Ω(n^1−є)), improving on the previous best lower bounds of Ω(√n) and exp(Ω(√n)), respectively. Our results subsume all previous lower bounds on the threshold degree and sign-rank of AC⁰ circuits of any given depth, with a strict improvement starting at depth 4. As a corollary, we also obtain near-optimal bounds on the discrepancy, threshold weight, and threshold density of AC⁰, strictly subsuming previous work on these quantities. Our work gives some of the strongest lower bounds to date on the communication complexity of AC⁰.

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https://dl.acm.org/doi/10.1145/3618260.3649776
Hatami HHosseini KMeng XSaha BServedio R(2023)A Borsuk-Ulam Lower Bound for Sign-Rank and Its ApplicationsProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585210(463-471)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585210
Sherstov ALeonardi SGupta A(2022)The approximate degree of DNF and CNF formulasProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520000(1194-1207)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519935.3520000
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Index Terms

Near-optimal lower bounds on the threshold degree and sign-rank of AC⁰
1. Theory of computation
  1. Computational complexity and cryptography

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cover image ACM Conferences

STOC 2019: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

June 2019

1258 pages

ISBN:9781450367059

DOI:10.1145/3313276

General Chair:
Moses Charikar
Stanford University
,
Program Chair:
Edith Cohen
Google, USA / Tel Aviv University, Israel

Copyright © 2019 ACM.

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Published: 23 June 2019

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STOC '19: 51st Annual ACM SIGACT Symposium on the Theory of Computing

June 23 - 26, 2019

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Cited By

Diakonikolas IKane DKontonis VLiu SZarifis NMohar BShinkar IO'Donnell R(2024)Super Non-singular Decompositions of Polynomials and Their Application to Robustly Learning Low-Degree PTFsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649776(152-159)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649776
Hatami HHosseini KMeng XSaha BServedio R(2023)A Borsuk-Ulam Lower Bound for Sign-Rank and Its ApplicationsProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585210(463-471)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585210
Sherstov ALeonardi SGupta A(2022)The approximate degree of DNF and CNF formulasProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520000(1194-1207)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519935.3520000
Karp SWinston ELi YSingh ARanzato MBeygelzimer ADauphin YLiang PVaughan J(2021)Local signal adaptivityProceedings of the 35th International Conference on Neural Information Processing Systems10.5555/3540261.3542167(24883-24897)Online publication date: 6-Dec-2021
https://dl.acm.org/doi/10.5555/3540261.3542167
Bun MThaler J(2021)Guest ColumnACM SIGACT News10.1145/3444815.344482551:4(48-72)Online publication date: 14-Jan-2021
https://dl.acm.org/doi/10.1145/3444815.3444825
Kretschmer W(2021)Lower Bounding the AND-OR Tree via SymmetrizationACM Transactions on Computation Theory10.1145/343438513:1(1-11)Online publication date: 21-Jan-2021
https://dl.acm.org/doi/10.1145/3434385

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