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Sparser Abelian High Dimensional Expanders

Yotam Dikstein, Siqi Liu, Avi Wigderson

Abstract

We present two new explicit constructions of Cayley high dimensional expanders (HDXs) over the abelian group $\mathbb{F}_2^n$. Our expansion proofs use only linear algebra and combinatorial arguments. The first construction gives local spectral HDXs of any constant dimension and subpolynomial degree $\exp(n^ε)$ for every $ε>0$, improving on a construction by Golowich [Gol23] which achieves $ε=1/2$. [Gol23] derives these HDXs by sparsifying the complete Grassmann poset of subspaces. The novelty in our construction is the ability to sparsify any expanding Grassmannian posets, leading to iterated sparsification and much smaller degrees. The sparse Grassmannian (which is of independent interest in the theory of HDXs) serves as the generating set of the Cayley graph. Our second construction gives a 2-dimensional HDXs of any polynomial degree $\exp(εn$) for any constant $ε> 0$, which is simultaneously a spectral expander and a coboundary expander. To the best of our knowledge, this is the first such non-trivial construction. We name it the Johnson complex, as it is derived from the classical Johnson scheme, whose vertices serve as the generating set of this Cayley graph. This construction may be viewed as a derandomization of the recent random geometric complexes of [LMSY23]. Establishing coboundary expansion through Gromov's "cone method" and the associated isoperimetric inequalities is the most intricate aspect of this construction. While these two constructions are quite different, we show that they both share a common structure, resembling the intersection patterns of vectors in the Hadamard code. We propose a general framework of such "Hadamard-like" constructions in the hope that it will yield new HDXs.

Sparser Abelian High Dimensional Expanders

Abstract

We present two new explicit constructions of Cayley high dimensional expanders (HDXs) over the abelian group . Our expansion proofs use only linear algebra and combinatorial arguments. The first construction gives local spectral HDXs of any constant dimension and subpolynomial degree for every , improving on a construction by Golowich [Gol23] which achieves . [Gol23] derives these HDXs by sparsifying the complete Grassmann poset of subspaces. The novelty in our construction is the ability to sparsify any expanding Grassmannian posets, leading to iterated sparsification and much smaller degrees. The sparse Grassmannian (which is of independent interest in the theory of HDXs) serves as the generating set of the Cayley graph. Our second construction gives a 2-dimensional HDXs of any polynomial degree ) for any constant , which is simultaneously a spectral expander and a coboundary expander. To the best of our knowledge, this is the first such non-trivial construction. We name it the Johnson complex, as it is derived from the classical Johnson scheme, whose vertices serve as the generating set of this Cayley graph. This construction may be viewed as a derandomization of the recent random geometric complexes of [LMSY23]. Establishing coboundary expansion through Gromov's "cone method" and the associated isoperimetric inequalities is the most intricate aspect of this construction. While these two constructions are quite different, we show that they both share a common structure, resembling the intersection patterns of vectors in the Hadamard code. We propose a general framework of such "Hadamard-like" constructions in the hope that it will yield new HDXs.

Paper Structure

This paper contains 86 sections, 45 theorems, 170 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Local spectral expanders
  3. Applications of local spectral expanders
  4. Previous constructions
  5. Our result
  6. Proof overview
  7. Our coboundary expanders
  8. Applications of coboundary expansion
  9. Known coboundary expanders
  10. Our result
  11. Cones and isoperimetry
  12. Proof overview
  13. Local spectral expansion and comparison to random geometric complexes
  14. The common structure between the two constructions
  15. Open questions
  16. ...and 71 more sections

Key Result

Theorem 1.2

For every $\lambda, \varepsilon > 0$ and integer $d \geqslant 2$ there exists an infinite family of $d$-dimensional Cayley $\lambda$-local spectral expanders over the vertices $\mathbb{F}_2^n$ with degree at most $2^{n^{\varepsilon}}$.

Figures (12)

  • Figure 1: A van Kampen diagram for $X,C_0$
  • Figure 2: The second item of prop:cone-like-isoperimetric-inequality
  • Figure 3: Two contraction patterns
  • Figure 4: Partition of $v_1 \cup v_1'$ in the proof of claim:four-cycle-in-X-prime
  • Figure 5: Half-step lexicographic cycle decomposition in lem:monotone-path-contraction
  • ...and 7 more figures

Theorems & Definitions (249)

  • Definition 1.1: Local spectral expansion
  • Theorem 1.2
  • Remark 1.3: Degree lower bounds for Cayley local spectral expanders
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6: Informal version of thm:coboundary-Johnson-complex, lem:Johnson-complex-spectral-expansion, and lem:cone-for-link-X
  • Remark 1.7
  • Definition 2.1: Expander graph
  • Definition 2.2: Bipartite expander graph
  • Claim 2.3
  • ...and 239 more