Constructing Bounded-Degree Symplectic High Dimensional Expanders with No Small Covers for Low Acceptance Agreement Tests
Основні поняття
Constructing bounded-degree symplectic high dimensional expanders that lack small covers, in order to enable low acceptance agreement tests.
Анотація
The paper presents a construction of new high dimensional expanders that have no small connected covers, and shows that these complexes support low acceptance agreement tests.
The key ideas are:
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Replace the well-studied LSV complexes, which are quotients of the SLn-affine building, with symplectic high dimensional expanders (HDXs) that are quotients of the affine Bruhat-Tits building associated with the symplectic group Sp(2g, Qp). These symplectic HDXs are shown to have no small covers, unlike the LSV complexes.
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Prove that the symplectic HDXs are swap cocycle expanders, which is a key requirement for the low acceptance agreement theorem to hold. This involves showing coboundary expansion and local spectral expansion for the color restrictions (subcomplexes) of the links of the symplectic building.
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Combine the lack of small covers and the swap cocycle expansion property to deduce a low acceptance agreement test theorem for the constructed symplectic HDXs, improving upon the previous best known results.
The paper also provides a polynomial-time algorithm to construct the family of symplectic HDXs used in the main theorem.
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Low Acceptance Agreement Tests via Bounded-Degree Symplectic HDXs
Статистика
The paper does not contain any specific numerical data or metrics to extract. The main results are theoretical in nature, providing constructions and proofs about the properties of high dimensional expanders.
Цитати
"We construct a family X of k-subsets of [N] such that |X| = O(N) and such that it satisfies the low acceptance agreement theorem."
"The family X is just the k-faces of the new symplectic HDXs. The later serve our needs better since their fundamental group satisfies the congruence subgroup property, which implies that they lack small covers."
Глибші Запити
Are there other desirable properties of the symplectic HDX complexes, beyond the lack of small covers, that could be leveraged for other applications in theoretical computer science
The symplectic HDX complexes possess several desirable properties beyond the lack of small covers that could be leveraged for various applications in theoretical computer science. One such property is their fundamental group satisfying the congruence subgroup property, which is crucial for the lack of small covers. Additionally, these complexes exhibit swap cocycle expansion, making them suitable for agreement tests and derandomized direct product testing. The high dimensional expanders derived from symplectic HDXs have been shown to support 1% agreement tests, indicating their robustness and efficiency in property testing scenarios. Furthermore, the construction of these complexes as quotients of the affine Bruhat-Tits building associated with the symplectic group Sp(2g, Qp) allows for a rich structure that can be exploited for various theoretical computer science applications.
Can the parameters in the low acceptance agreement theorem be further improved by using different building-based constructions of high dimensional expanders, beyond the symplectic and SLn cases
While the symplectic and SLn cases have been instrumental in advancing the parameters in the low acceptance agreement theorem, there is potential for further improvement by exploring different building-based constructions of high dimensional expanders. By considering alternative constructions and leveraging different types of buildings, it may be possible to enhance the trade-off between set size, degree, and soundness in agreement tests. Exploring new types of high dimensional expanders and their properties could lead to advancements in the parameters of the agreement theorem, potentially achieving even better results in terms of efficiency and accuracy in property testing scenarios.
Are there bounded-degree local spectral expanders that are coboundary expanders with respect to all possible groups, not just finite groups
The existence of bounded-degree local spectral expanders that are coboundary expanders with respect to all possible groups, not just finite groups, remains an intriguing open problem in this area. Constructing such complexes would require a deep understanding of the spectral properties and structural characteristics that enable coboundary expansion with respect to a wide range of groups. This endeavor presents a challenging yet rewarding opportunity to explore the connections between spectral expansion, group theory, and combinatorial structures, potentially leading to novel insights and advancements in the field of theoretical computer science.
