Towards a calculus for non-linear spectral gaps

Manor Mendel, Assaf Naor

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Given a finite regular graph G = (V, E) and a metric space (X, d X), let γ+(G, X) denote the smallest constant γ+ > 0 such that for all f, g : V → X we have: 1/|V|2 ∑ x,y∈V dx(f(x),g(y))2 ≤ γ+/|E| ∑ xy∈E dx(f(x),g(y))2. In the special case X = R this quantity coincides with the reciprocal of the absolute spectral gap of G, but for other geometries the parameter γ+(G, X), which we still think of as measuring the non-linear spectral gap of G with respect to X (even though there is no actual spectrum present here), can behave very differently. Non-linear spectral gaps arise often in the theory of metric embeddings, and in the present paper we systematically study the theory of non-linear spectral gaps, partially in order to obtain a combinatorial construction of super-expander - a family of bounded-degree graphs Gi = (V i, Ei), with limi→∞ |Vi| = ∞, which do not admit a coarse embedding into any uniformly convex normed space. In addition, the bi-Lipschitz distortion of Gi in any uniformly convex Banach space is Ω(log |Vi|), which is the worst possible behavior due to Bourgain's embedding theorem [3]. Such remarkable graph families were previously known to exist due to a tour de force algebraic construction of Lafforgue [11]. Our construction is different and combinatorial, relying on the zigzag product of Reingold-Vadhan-Wigderson [28]. We show that non-linear spectral gaps behave submultiplicatively under zigzag products - a fact that amounts to a simple iteration of the inequality above. This yields as a special case a very simple (linear algebra free) proof of the Reingold-Vadhan-Wigderson theorem which states that zigzag products preserve the property of having an absolute spectral gap (with quantitative control on the size of the gap). The zigzag iteration of Reingold-Vadhan-Wigderson also involves taking graph powers, which is trivial to analyze in the classical "linear" setting. In our work, the behavior of non-linear spectral gaps under graph powers becomes a major geometric obstacle, and we show that for uniformly convex normed spaces there exists a satisfactory substitute for spectral calculus which makes sense in the non-linear setting. These facts, in conjunction with a variant of Ball's notion of Markov cotype and a Fourier analytic proof of the existence of appropriate "base graphs", are shown to imply that Reingold-Vadhan-Wigderson type constructions can be carried out in the non-linear setting.

Original languageEnglish (US)
Title of host publicationProceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms
Pages236-255
Number of pages20
StatePublished - 2010
Event21st Annual ACM-SIAM Symposium on Discrete Algorithms - Austin, TX, United States
Duration: Jan 17 2010Jan 19 2010

Other

Other21st Annual ACM-SIAM Symposium on Discrete Algorithms
CountryUnited States
CityAustin, TX
Period1/17/101/19/10

Fingerprint

Spectral Gap
Calculus
Zigzag
Uniformly Convex Space
Graph Powers
Normed Space
Linear algebra
Banach spaces
Graph in graph theory
Metric Embeddings
Iteration
Expander
Uniformly Convex Banach Space
Embedding Theorem
Finite Graph
Geometry
Substitute
Regular Graph
Metric space
Lipschitz

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

Cite this

Mendel, M., & Naor, A. (2010). Towards a calculus for non-linear spectral gaps. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 236-255)

Towards a calculus for non-linear spectral gaps. / Mendel, Manor; Naor, Assaf.

Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms. 2010. p. 236-255.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Mendel, M & Naor, A 2010, Towards a calculus for non-linear spectral gaps. in Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms. pp. 236-255, 21st Annual ACM-SIAM Symposium on Discrete Algorithms, Austin, TX, United States, 1/17/10.
Mendel M, Naor A. Towards a calculus for non-linear spectral gaps. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms. 2010. p. 236-255
Mendel, Manor ; Naor, Assaf. / Towards a calculus for non-linear spectral gaps. Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms. 2010. pp. 236-255
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