### 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 G_{i} = (V _{i}, E_{i}), with lim_{i→∞} |V_{i}| = ∞, which do not admit a coarse embedding into any uniformly convex normed space. In addition, the bi-Lipschitz distortion of G_{i} in any uniformly convex Banach space is Ω(log |V_{i}|), 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 language | English (US) |
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Title of host publication | Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms |

Pages | 236-255 |

Number of pages | 20 |

State | Published - 2010 |

Event | 21st Annual ACM-SIAM Symposium on Discrete Algorithms - Austin, TX, United States Duration: Jan 17 2010 → Jan 19 2010 |

### Other

Other | 21st Annual ACM-SIAM Symposium on Discrete Algorithms |
---|---|

Country | United States |

City | Austin, TX |

Period | 1/17/10 → 1/19/10 |

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### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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.

}

TY - GEN

T1 - Towards a calculus for non-linear spectral gaps

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77951674750&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951674750&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9780898717013

SP - 236

EP - 255

BT - Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms

ER -