Hypergraph expanders from Cayley graphs

We present a simple mechanism, which can be randomised, for constructing sparse $3$-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over $\mathbb{Z}_2^t$ and have vertex degree which is polylogarithmic in the number of vertices. Their expansion properties, which are derived from the underlying Cayley graphs, include analogues of vertex and edge expansion in graphs, rapid mixing of the random walk on the edges of the skeleton graph, uniform distribution of edges on large vertex subsets and the geometric overlap property.Comment: 13 page

Unique-Neighbor-Like Expansion and Group-Independent Cosystolic Expansion

In recent years, high dimensional expanders have been found to have a variety of applications in theoretical computer science, such as efficient CSPs approximations, improved sampling and list-decoding algorithms, and more. Within that, an important high dimensional expansion notion is cosystolic expansion, which has found applications in the construction of efficiently decodable quantum codes and in proving lower bounds for CSPs. Cosystolic expansion is considered with systems of equations over a group where the variables and equations correspond to faces of the complex. Previous works that studied cosystolic expansion were tailored to the specific group ??. In particular, Kaufman, Kazhdan and Lubotzky (FOCS 2014), and Evra and Kaufman (STOC 2016) in their breakthrough works, who solved a famous open question of Gromov, have studied a notion which we term "parity" expansion for small sets. They showed that small sets of k-faces have proportionally many (k+1)-faces that contain an odd number of k-faces from the set. Parity expansion for small sets could then be used to imply cosystolic expansion only over ??. In this work we introduce a stronger unique-neighbor-like expansion for small sets. We show that small sets of k-faces have proportionally many (k+1)-faces that contain exactly one k-face from the set. This notion is fundamentally stronger than parity expansion and cannot be implied by previous works. We then show, utilizing the new unique-neighbor-like expansion notion introduced in this work, that cosystolic expansion can be made group-independent, i.e., unique-neighbor-like expansion for small sets implies cosystolic expansion over any group

Coboundary and cosystolic expansion without dependence on dimension or degree

We give new bounds on the cosystolic expansion constants of several families of high dimensional expanders, and the known coboundary expansion constants of order complexes of homogeneous geometric lattices, including the spherical building of $SL_n(F_q)$. The improvement applies to the high dimensional expanders constructed by Lubotzky, Samuels and Vishne, and by Kaufman and Oppenheim. Our new expansion constants do not depend on the degree of the complex nor on its dimension, nor on the group of coefficients. This implies improved bounds on Gromov's topological overlap constant, and on Dinur and Meshulam's cover stability, which may have applications for agreement testing. In comparison, existing bounds decay exponentially with the ambient dimension (for spherical buildings) and in addition decay linearly with the degree (for all known bounded-degree high dimensional expanders). Our results are based on several new techniques: * We develop a new "color-restriction" technique which enables proving dimension-free expansion by restricting a multi-partite complex to small random subsets of its color classes. * We give a new "spectral" proof for Evra and Kaufman's local-to-global theorem, deriving better bounds and getting rid of the dependence on the degree. This theorem bounds the cosystolic expansion of a complex using coboundary expansion and spectral expansion of the links. * We derive absolute bounds on the coboundary expansion of the spherical building (and any order complex of a homogeneous geometric lattice) by constructing a novel family of very short cones

Couboundary Expansion of Sheaves on Graphs and Weighted Mixing Lemmas

We study the coboundary expansion of graphs, but instead of using $\mathbb{F}_2$ as the coefficient group when forming the cohomology, we use a sheaf on the graph. We prove that if the graph under discussion is a good expander, then it is also a good coboundary expander relative to any constant augmented sheaf (equivalently, relative to any coefficient group $R$); this, however, may fail for locally constant sheaves. We moreover show that if we take the quotient of a constant augmented sheaf on an excellent expander graph by a "small" subsheaf, then the quotient sheaf is still a good coboundary expander. Along the way, we prove a new version of the Expander Mixing Lemma applying to $r$-partite weighted graphs.Comment: Comments are welcom

Coboundary expansion for the union of determinantal hypertrees

We prove that for any large enough constant $k$, the union of $k$ independent $d$-dimensional determinantal hypertrees is a coboundary expander with high probability

High Dimensional Expansion Implies Amplified Local Testability

In this work, we define a notion of local testability of codes that is strictly stronger than the basic one (studied e.g., by recent works on high rate LTCs), and we term it amplified local testability. Amplified local testability is a notion close to the result of optimal testing for Reed-Muller codes achieved by Bhattacharyya et al. We present a scheme to get amplified locally testable codes from high dimensional expanders. We show that single orbit Affine invariant codes, and in particular Reed-Muller codes, can be described via our scheme, and hence are amplified locally testable. This gives the strongest currently known testability result of single orbit affine invariant codes, strengthening the celebrated result of Kaufman and Sudan