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Almost Ramanujan Expanders from Arbitrary Expanders via Operator Amplification

Fernando Granha Jeronimo, Tushant Mittal, Sourya Roy, Avi Wigderson

TL;DR

An efficient algorithm that transforms any bounded degree expander graph into another that achieves almost optimal trade-off between (any desired) spectral expansion $\lambda$ and degree d, and extends Ta-Shma’s analysis of bias amplification from scalars to matrices of arbitrary dimension in a very natural way.

Abstract

We give an efficient algorithm that transforms any bounded degree expander graph into another that achieves almost optimal (namely, near-quadratic, $d \leq 1/λ^{2+o(1)}$) trade-off between (any desired) spectral expansion $λ$ and degree $d$. Furthermore, the algorithm is local: every vertex can compute its new neighbors as a subset of its original neighborhood of radius $O(\log(1/λ))$. The optimal quadratic trade-off is known as the Ramanujan bound, so our construction gives almost Ramanujan expanders from arbitrary expanders. The locality of the transformation preserves structural properties of the original graph, and thus has many consequences. Applied to Cayley graphs, our transformation shows that any expanding finite group has almost Ramanujan expanding generators. Similarly, one can obtain almost optimal explicit constructions of quantum expanders, dimension expanders, monotone expanders, etc., from existing (suboptimal) constructions of such objects. Another consequence is a "derandomized" random walk on the original (suboptimal) expander with almost optimal convergence rate. Our transformation also applies when the degree is not bounded or the expansion is not constant. We obtain our results by a generalization of Ta-Shma's technique in his breakthrough paper [STOC 2017], used to obtain explicit almost optimal binary codes. Specifically, our spectral amplification extends Ta-Shma's analysis of bias amplification from scalars to matrices of arbitrary dimension in a very natural way. Curiously, while Ta-Shma's explicit bias amplification derandomizes a well-known probabilistic argument (underlying the Gilbert--Varshamov bound), there seems to be no known probabilistic (or other existential) way of achieving our explicit ("high-dimensional") spectral amplification.

Almost Ramanujan Expanders from Arbitrary Expanders via Operator Amplification

TL;DR

An efficient algorithm that transforms any bounded degree expander graph into another that achieves almost optimal trade-off between (any desired) spectral expansion and degree d, and extends Ta-Shma’s analysis of bias amplification from scalars to matrices of arbitrary dimension in a very natural way.

Abstract

We give an efficient algorithm that transforms any bounded degree expander graph into another that achieves almost optimal (namely, near-quadratic, ) trade-off between (any desired) spectral expansion and degree . Furthermore, the algorithm is local: every vertex can compute its new neighbors as a subset of its original neighborhood of radius . The optimal quadratic trade-off is known as the Ramanujan bound, so our construction gives almost Ramanujan expanders from arbitrary expanders. The locality of the transformation preserves structural properties of the original graph, and thus has many consequences. Applied to Cayley graphs, our transformation shows that any expanding finite group has almost Ramanujan expanding generators. Similarly, one can obtain almost optimal explicit constructions of quantum expanders, dimension expanders, monotone expanders, etc., from existing (suboptimal) constructions of such objects. Another consequence is a "derandomized" random walk on the original (suboptimal) expander with almost optimal convergence rate. Our transformation also applies when the degree is not bounded or the expansion is not constant. We obtain our results by a generalization of Ta-Shma's technique in his breakthrough paper [STOC 2017], used to obtain explicit almost optimal binary codes. Specifically, our spectral amplification extends Ta-Shma's analysis of bias amplification from scalars to matrices of arbitrary dimension in a very natural way. Curiously, while Ta-Shma's explicit bias amplification derandomizes a well-known probabilistic argument (underlying the Gilbert--Varshamov bound), there seems to be no known probabilistic (or other existential) way of achieving our explicit ("high-dimensional") spectral amplification.
Paper Structure (41 sections, 46 theorems, 41 equations)

This paper contains 41 sections, 46 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Results
  4. Cayley Graphs
  5. Applications
  6. Techniques
  7. A first approach
  8. Scalar amplification
  9. Operator amplification
  10. Our Results
  11. Discussion
  12. Dimension Independence
  13. Algebraic vs. Combinatorial Expander Constructions
  14. Iterated Pseudorandomness
  15. Groups and Expansion
  16. ...and 26 more sections

Key Result

Theorem 1.1

Let $\lbrace X_i\rbrace_{i \in \mathbb{N}}$ be a family of $(d_0,\lambda_0)$-expanders where $\lambda_0 < 1$ is a constant. For any (target) $\lambda \in (0,1)$ and $X_i$, we can explicitlySee def:explicit construct a $(d,\lambda)$-expander, $X_i'$, on the same vertex set, where $d = O(d_0/\lambda^{

Theorems & Definitions (71)

  • Theorem 1.1: Main I - Informal
  • Theorem 1.2: Main II
  • Corollary 1.3: Explicit almost Ramanujan Cayley Expanders
  • Remark 1.4: Connection with Codes
  • Corollary 1.4: Explicit Almost Ramanujan Quantum Expanders
  • Corollary 1.4: Almost Ramanujan Monotone Expanders
  • Corollary 1.4: Amplifying Average Kazhdan Constant
  • Theorem 1.5: Template Amplification Result
  • Theorem 1.6: Operator Amplification (this work)
  • Definition 2.1: $\lambda$-spectral Expander
  • ...and 61 more