insight - Computational Complexity - # Agreement Testing on High Dimensional Expanders

Agreement Theorems for High Dimensional Expanders in the Low Acceptance Regime: The Role of Topological Covers

Q: Can the requirement of swap-cosystolic-expansion be weakened further while still obtaining agreement theorems

In the context of high dimensional expanders and agreement theorems, the requirement of swap-cosystolic-expansion can potentially be weakened further while still obtaining agreement theorems. The swap-cosystolic-expansion condition plays a crucial role in the main theorem presented in the context. It ensures that the complex satisfies certain properties that allow for the derivation of agreement theorems in the low acceptance regime. To weaken this requirement, one could explore alternative conditions or properties of the complex that still facilitate the construction of agreement theorems. This could involve investigating different spectral properties, connectivity structures, or other characteristics of the complex that contribute to the ability to derive agreement theorems. By identifying and utilizing these alternative conditions, it may be possible to relax the swap-cosystolic-expansion requirement while still achieving the desired results in terms of agreement theorems.

Q: What are the limits of derandomization for agreement tests

The limits of derandomization for agreement tests are closely tied to the construction of linear-size families of complexes that satisfy the classical 1% agreement theorem. In the context provided, the goal is to find families of complexes that support agreement theorems in the low acceptance regime while maintaining a linear size. Derandomization in this context involves finding structured families of complexes that exhibit certain properties allowing for agreement theorems to be derived without relying on randomness. The construction of linear-size families satisfying the classical 1% agreement theorem is a significant challenge but holds promise for advancing the understanding of agreement tests in high dimensional expanders. By exploring different constructions, properties, and techniques, it may be possible to push the limits of derandomization further and potentially achieve linear-size families of complexes that satisfy the classical 1% agreement theorem. This would represent a significant advancement in the field of agreement theorems for high dimensional expanders.

Q: Can linear-size families of complexes satisfying the classical 1% agreement theorem be constructed

The parameters obtained in the agreement theorems presented in the context compared to the classical results on the complete complex showcase advancements and potential improvements in the field. The dependence on ε, which represents the acceptance threshold in the low acceptance regime, is a critical factor in determining the strength and applicability of the agreement theorems. In the classical results on the complete complex, the parameters for agreement theorems in the low acceptance regime are well-studied and understood. The results obtained in the context, particularly in the main theorem, demonstrate advancements in terms of the parameters, such as the dependence on ε. By achieving results with ε = Ω(1/ log k), the agreement theorems presented in the context show potential for improved performance and applicability in practical scenarios. Overall, the comparison of parameters between the classical results and the results obtained in the context highlights progress and potential enhancements in the field of agreement theorems for high dimensional expanders.

Conceitos Básicos

Even sparse high dimensional expanders may fail to satisfy 1% agreement theorems due to the presence of connected topological covers. However, a weaker agreement theorem can still be obtained by taking the cover structure into account.

Resumo

The paper studies agreement theorems for high dimensional expanders in the low acceptance (or 1%) regime. Agreement tests are used to determine if an ensemble of local functions {fs : s → Σ | s ∈ X} on a simplicial complex X can be "explained" by a global function G : [n] → Σ.

The main results are:

If the complex X has no connected covers, then the classical 1% agreement theorem holds, provided X satisfies an additional property called swap cosystolic expansion.
If X has a connected cover, then the classical 1% agreement theorem fails.
If X has a connected cover (and satisfies swap-cosystolic-expansion), a weaker agreement theorem is shown to hold. This theorem takes the cover structure into account, showing that the ensemble {fs} can be "explained" by a global function G defined on a cover Y of X.

The paper also shows that many known constructions of sparse high dimensional expanders, including spherical buildings and LSV complexes, satisfy the required properties to obtain these agreement theorems. This improves upon previous results, providing the sparsest known families of complexes that support 1% agreement theorems.

The technical approach involves constructing compatible lists of local functions on the faces of the complex, using the expansion properties to resolve local inconsistencies, and then lifting these to a global function on a cover of the complex.

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Estatísticas

Agree({fs}) > ε =⇒ ∃cover ρ : Y ↠ X, and G : Y (0) → Σ, such that P[fs 0.99≈ G|s̃] ≥ poly(ε), where s̃ ↠ s means that ρ(s̃) = s.

Citações

"If X has a connected cover, then (LA) fails."
"If X has a connected cover (and swap-cosystolic-expansion), we replace (LA) by a statement that takes covers into account."

Principais Insights Extraídos De

Agreement theorems for high dimensional expanders in the small soundness regime: the role of covers

by Yotam Dikste... às arxiv.org 04-15-2024

https://arxiv.org/pdf/2308.09582.pdf

Agreement theorems for high dimensional expanders in the small soundness regime: the role of covers

Perguntas Mais Profundas

Can the requirement of swap-cosystolic-expansion be weakened further while still obtaining agreement theorems

In the context of high dimensional expanders and agreement theorems, the requirement of swap-cosystolic-expansion can potentially be weakened further while still obtaining agreement theorems. The swap-cosystolic-expansion condition plays a crucial role in the main theorem presented in the context. It ensures that the complex satisfies certain properties that allow for the derivation of agreement theorems in the low acceptance regime.
To weaken this requirement, one could explore alternative conditions or properties of the complex that still facilitate the construction of agreement theorems. This could involve investigating different spectral properties, connectivity structures, or other characteristics of the complex that contribute to the ability to derive agreement theorems. By identifying and utilizing these alternative conditions, it may be possible to relax the swap-cosystolic-expansion requirement while still achieving the desired results in terms of agreement theorems.

What are the limits of derandomization for agreement tests

The limits of derandomization for agreement tests are closely tied to the construction of linear-size families of complexes that satisfy the classical 1% agreement theorem. In the context provided, the goal is to find families of complexes that support agreement theorems in the low acceptance regime while maintaining a linear size.
Derandomization in this context involves finding structured families of complexes that exhibit certain properties allowing for agreement theorems to be derived without relying on randomness. The construction of linear-size families satisfying the classical 1% agreement theorem is a significant challenge but holds promise for advancing the understanding of agreement tests in high dimensional expanders.
By exploring different constructions, properties, and techniques, it may be possible to push the limits of derandomization further and potentially achieve linear-size families of complexes that satisfy the classical 1% agreement theorem. This would represent a significant advancement in the field of agreement theorems for high dimensional expanders.

Can linear-size families of complexes satisfying the classical 1% agreement theorem be constructed

The parameters obtained in the agreement theorems presented in the context compared to the classical results on the complete complex showcase advancements and potential improvements in the field. The dependence on ε, which represents the acceptance threshold in the low acceptance regime, is a critical factor in determining the strength and applicability of the agreement theorems.
In the classical results on the complete complex, the parameters for agreement theorems in the low acceptance regime are well-studied and understood. The results obtained in the context, particularly in the main theorem, demonstrate advancements in terms of the parameters, such as the dependence on ε. By achieving results with ε = Ω(1/ log k), the agreement theorems presented in the context show potential for improved performance and applicability in practical scenarios.
Overall, the comparison of parameters between the classical results and the results obtained in the context highlights progress and potential enhancements in the field of agreement theorems for high dimensional expanders.