Characterizing Direct Product Testing via Coboundary Expansion
Name
3618260.3649714.pdf
Size
301.75 KB
Format
Adobe PDF
Checksum (MD5)
b2d9dd67b128fc174ce6103e2ac9208d
Author(s)
Bafna, Mitali
Minzer, Dor
Date Issued
June 10, 2024
Publisher
ACM
Citation
Bafna, Mitali and Minzer, Dor. 2024. "Characterizing Direct Product Testing via Coboundary Expansion."
Version
Final published version
Abstract
A d-dimensional simplicial complex X is said to support a direct product tester if any locally consistent function defined on its k-faces (where k≪ d) necessarily come from a function over its vertices. More precisely, a direct product tester has a distribution µ over pairs of k-faces (A,A′), and given query access to F: X(k)→{0,1}k it samples (A,A′)∼ µ and checks that F[A]|A∩ A′ = F[A′]|A∩ A′. The tester should have (1) the ”completeness property”, meaning that any assignment F which is a direct product assignment passes the test with probability 1, and (2) the ”soundness property”, meaning that if F passes the test with probability s, then F must be correlated with a direct product function. Dinur and Kaufman showed that a sufficiently good spectral expanding complex X admits a direct product tester in the ”high soundness” regime where s is close to 1. They asked whether there are high dimensional expanders that support direct product tests in the ”low soundness”, when s is close to 0. We give a characterization of high-dimensional expanders that support a direct product tester in the low soundness regime. We show that spectral expansion is insufficient, and the complex must additionally satisfy a variant of coboundary expansion, which we refer to as ”Unique-Games coboundary expanders”. Conversely, we show that this property is also sufficient to get direct product testers. This property can be seen as a high-dimensional generalization of the standard notion of coboundary expansion over non-Abelian groups for 2-dimensional complexes. It asserts that any locally consistent Unique-Games instance obtained using the low-level faces of the complex, must admit a good global solution.
Description
STOC ’24, June 24–28, 2024, Vancouver, BC, Canada
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1145/3618260.3649714
