arXiv subjects

Nathan Linial

Publications and source records attributed to Nathan Linial.

22 records · Page 2Linked to original sources

arXiv2004

Limitations to Frechet's Metric Embedding Method

Frechet's classical isometric embedding argument has evolved to become a major tool in the study of metric spaces. An important example of a Frechet embedding is Bourgain's embedding. The authors have recently shown that for every e>0 any n-point metric space contains a subset of size at least n^(1-e) which embeds into l_2 with distortion O(\log(2/e) /e). The embedding we used is non-Frechet, and the purpose of this note is to show that this is not coincidental. Specifically, for every e>0, we construct arbitrarily large n-point metric spaces, such that the distortion of any Frechet embedding into l_p on subsets of size at least n^{1/2 + e} is Ω((\log n)^{1/p}).

math.MG↗

arXiv2004

On Metric Ramsey-type Dichotomies

The classical Ramsey theorem, states that every graph contains either a large clique or a large independent set. Here we investigate similar dichotomic phenomena in the context of finite metric spaces. Namely, we prove statements of the form "Every finite metric space contains a large subspace that is nearly quilateral or far from being equilateral". We consider two distinct interpretations for being "far from equilateral". Proximity among metric spaces is quantified through the metric distortion D. We provide tight asymptotic answers for these problems. In particular, we show that a phase transition occurs at D=2.

math.CO↗

arXiv2004

Constructing expander graphs by 2-lifts and discrepancy vs. spectral gap

We present a new explicit construction for expander graphs with nearly optimal spectral gap. The construction is based on a series of 2-lift operations. Let $G$ be a graph on $n$ vertices. A 2-lift of $G$ is a graph $H$ on $2n$ vertices, with a covering map $π:H \to G$. It is not hard to see that all eigenvalues of $G$ are also eigenvalues of $H$. In addition, $H$ has $n$ ``new'' eigenvalues. We conjecture that every $d$-regular graph has a 2-lift such that all new eigenvalues are in the range $[-2\sqrt{d-1},2\sqrt{d-1}]$ (If true, this is tight, e.g. by the Alon-Boppana bound). Here we show that every graph of maximal degree $d$ has a 2-lift such that all ``new'' eigenvalues are in the range $[-c \sqrt{d \log^3d}, c \sqrt{d \log^3d}]$ for some constant $c$. This leads to a polynomial time algorithm for constructing arbitrarily large $d$-regular graphs, with second eigenvalue $O(\sqrt{d \log^3 d})$. The proof uses the following lemma: Let $A$ be a real symmetric matrix such that the $l_1$ norm of each row in $A$ is at most $d$. Let $α= \max_{x,y \in \{0,1\}^n, supp(x)\cap supp(y)=\emptyset} \frac {|xAy|} {||x||||y||}$. Then the spectral radius of $A$ is at most $c α\log(d/α)$, for some universal constant $c$. An interesting consequence of this lemma is a converse to the Expander Mixing Lemma.

math.CO↗

arXiv2003

Finite metric spaces--combinatorics, geometry and algorithms

Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information. Metric spaces also come up in many recent advances in the theory of algorithms. Finally, finite submetrics of classical geometric objects such as normed spaces or manifolds reflect many important properties of the underlying structure. In this paper we review some of the recent advances in this area.

math.CO↗