Expanders with respect to Hadamard spaces and random graphs
Manor Mendel,Assaf Naor +1 more
TLDR
The Euclidean cone over a random graph is used as an auxiliary continuous geometric object that allows for the implementation of martingale methods, yielding a deterministic sublinear time constant factor approximation algorithm for computing the average squared distance in subsets of a random graphs.Abstract:
It is shown that there exist a sequence of 3 -regular graphs { G n } n = 1 ∞ and a Hadamard space X such that { G n } n = 1 ∞ forms an expander sequence with respect to X , yet random regular graphs are not expanders with respect to X . This answers a question of the second author and Silberman. The graphs { G n } n = 1 ∞ are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear-time constant-factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of martingale methods.read more
Citations
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Book
Convex analysis and optimization in Hadamard spaces
TL;DR: In this article, the authors give a systematic account on the subject of convex analysis and optimization in Hadamard spaces, aimed at both graduate students and researchers in analysis and optimisation.
Journal ArticleDOI
Nonlinear spectral calculus and super-expanders
Manor Mendel,Assaf Naor +1 more
TL;DR: In this paper, nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages, which yields a combinatorial construction of superexpanders.
Proceedings ArticleDOI
Data-dependent hashing via nonlinear spectral gaps
TL;DR: A generic reduction from _nonlinear spectral gaps_ of metric spaces to data-dependent Locality-Sensitive Hashing is established, yielding a new approach to the high-dimensional Approximate Near Neighbor Search problem (ANN) under various distance functions.
Journal ArticleDOI
Spectral calculus and Lipschitz extension for barycentric metric spaces
Manor Mendel,Assaf Naor +1 more
TL;DR: In this article, the metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this bi-Lipschitz invariant is understood.
Journal ArticleDOI
On Lipschitz extension from finite subsets
Assaf Naor,Yuval Rabani +1 more
TL;DR: In this paper, it was shown that for every n ∈ ℕ there exists a metric space (X, d ≥ 0, X), an n-point subset S ⊆ X, a Banach space (Z, $${\left\| \right\|_Z}$$ ), and a 1-Lipschitz function f: S → Z such that the Lipschitzer constant of every function F: X → Z that extends f is at least a constant multiple of $$\sqrt {\log n} $$.
References
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Book
The Probabilistic Method
TL;DR: A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.
Book
Metric Spaces of Non-Positive Curvature
TL;DR: In this article, the authors describe the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by isometries.
Journal ArticleDOI
Expander graphs and their applications
S Hoory,Nathan Linial +1 more
TL;DR: Expander graphs were first defined by Bassalygo and Pinsker in the early 1970s, and their existence was proved in the late 1970s as discussed by the authors and early 1980s.
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The geometry of graphs and some of its algorithmic applications
TL;DR: Efficient algorithms for embedding graphs low-dimensionally with a small distortion, and a new deterministic polynomial-time algorithm that finds a (nearly tight) cut meeting this bound.