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Assaf Naor

Publications and source records attributed to Assaf Naor.

At least 91 records · Page 5Linked to original sources
arXiv2009

Towards a Calculus for Non-Linear Spectral Gaps [Extended Abstract]

Given a finite regular graph G=(V,E) and a metric space (X,d_X), let $gamma_+(G,X) denote the smallest constant $γ_+>0$ such that for all f,g:V\to X we have: \frac{1}{|V|^2}\sum_{x,y\in V} d_X(f(x),g(y))^2\le \frac{γ_+}{|E|} \sum_{xy\in E} d_X(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\to \infty} |V_i|=\infty, 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. Such remarkable graph families were previously known to exist due to a tour de force algebraic construction of Lafforgue. Our construction is different and combinatorial, relying on the zigzag product of Reingold-Vadhan-Wigderson.

math.MG↗
arXiv2009

Sharp kernel clustering algorithms and their associated Grothendieck inequalities

In the kernel clustering problem we are given a (large) $n\times n$ symmetric positive semidefinite matrix $A=(a_{ij})$ with $\sum_{i=1}^n\sum_{j=1}^n a_{ij}=0$ and a (small) $k\times k$ symmetric positive semidefinite matrix $B=(b_{ij})$. The goal is to find a partition $\{S_1,...,S_k\}$ of $\{1,... n\}$ which maximizes $ \sum_{i=1}^k\sum_{j=1}^k (\sum_{(p,q)\in S_i\times S_j}a_{pq})b_{ij}$. We design a polynomial time approximation algorithm that achieves an approximation ratio of $\frac{R(B)^2}{C(B)}$, where $R(B)$ and $C(B)$ are geometric parameters that depend only on the matrix $B$, defined as follows: if $b_{ij} = < v_i, v_j>$ is the Gram matrix representation of $B$ for some $v_1,...,v_k\in \R^k$ then $R(B)$ is the minimum radius of a Euclidean ball containing the points $\{v_1, ..., v_k\}$. The parameter $C(B)$ is defined as the maximum over all measurable partitions $\{A_1,...,A_k\}$ of $\R^{k-1}$ of the quantity $\sum_{i=1}^k\sum_{j=1}^k b_{ij}< z_i,z_j>$, where for $i\in \{1,...,k\}$ the vector $z_i\in \R^{k-1}$ is the Gaussian moment of $A_i$, i.e., $z_i=\frac{1}{(2π)^{(k-1)/2}}\int_{A_i}xe^{-\|x\|_2^2/2}dx$. We also show that for every $\eps > 0$, achieving an approximation guarantee of $(1-\e)\frac{R(B)^2}{C(B)}$ is Unique Games hard.

cs.DS↗
arXiv2009

$L_p$ compression, traveling salesmen, and stable walks

We show that if $H$ is a group of polynomial growth whose growth rate is at least quadratic then the $L_p$ compression of the wreath product $\Z\bwr H$ equals $\max{\frac{1}{p},{1/2}}$. We also show that the $L_p$ compression of $\Z\bwr \Z$ equals $\max{\frac{p}{2p-1},\frac23}$ and the $L_p$ compression of $(\Z\bwr\Z)_0$ (the zero section of $\Z\bwr \Z$, equipped with the metric induced from $\Z\bwr \Z$) equals $\max{\frac{p+1}{2p},\frac34}$. The fact that the Hilbert compression exponent of $\Z\bwr\Z$ equals $\frac23$ while the Hilbert compression exponent of $(\Z\bwr\Z)_0$ equals $\frac34$ is used to show that there exists a Lipschitz function $f:(\Z\bwr\Z)_0\to L_2$ which cannot be extended to a Lipschitz function defined on all of $\Z\bwr \Z$.

math.MG↗
arXiv2008

Approximate kernel clustering

In the kernel clustering problem we are given a large $n\times n$ positive semi-definite matrix $A=(a_{ij})$ with $\sum_{i,j=1}^na_{ij}=0$ and a small $k\times k$ positive semi-definite matrix $B=(b_{ij})$. The goal is to find a partition $S_1,...,S_k$ of $\{1,... n\}$ which maximizes the quantity $$ \sum_{i,j=1}^k (\sum_{(i,j)\in S_i\times S_j}a_{ij})b_{ij}. $$ We study the computational complexity of this generic clustering problem which originates in the theory of machine learning. We design a constant factor polynomial time approximation algorithm for this problem, answering a question posed by Song, Smola, Gretton and Borgwardt. In some cases we manage to compute the sharp approximation threshold for this problem assuming the Unique Games Conjecture (UGC). In particular, when $B$ is the $3\times 3$ identity matrix the UGC hardness threshold of this problem is exactly $\frac{16π}{27}$. We present and study a geometric conjecture of independent interest which we show would imply that the UGC threshold when $B$ is the $k\times k$ identity matrix is $\frac{8π}{9}(1-\frac{1}{k})$ for every $k\ge 3$.

cs.DS↗
arXiv2008

The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite

Let $X$ be a normed space that satisfies the Johnson-Lindenstrauss lemma (J-L lemma, in short) in the sense that for any integer $n$ and any $x_1,\ldots,x_n\in X$ there exists a linear mapping $L:X\to F$, where $F\subseteq X$ is a linear subspace of dimension $O(\log n)$, such that $\|x_i-x_j\|\le\|L(x_i)-L(x_j)\|\le O(1)\cdot\|x_i-x_j\|$ for all $i,j\in \{1,\ldots, n\}$. We show that this implies that $X$ is almost Euclidean in the following sense: Every $n$-dimensional subspace of $X$ embeds into Hilbert space with distortion $2^{2^{O(\log^*n)}}$. On the other hand, we show that there exists a normed space $Y$ which satisfies the J-L lemma, but for every $n$ there exists an $n$-dimensional subspace $E_n\subseteq Y$ whose Euclidean distortion is at least $2^{Ω(α(n))}$, where $α$ is the inverse Ackermann function.

math.FA↗
arXiv2007

Embeddings of discrete groups and the speed of random walks

For a finitely generated group G and a banach space X let α^*_X(G) (respectively α^#_X(G)) be the supremum over all α\ge 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f:G\to X and c>0 such that for all x,y\in G we have \|f(x)-f(y)\|\ge c\cdot d_G(x,y)^α. In particular, the Hilbert compression exponent (respectively the equivariant Hilbert compression exponent) of G is α^*(G)=α^*_{L_2}(G) (respectively α^#(G)= α_{L_2}^#(G)). We show that if X has modulus of smoothness of power type p, then α^#_X(G)\le \frac{1}{pβ^*(G)}. Here β^*(G) is the largest β\ge 0 for which there exists a set of generators S of G and c>0 such that for all t\in \N we have \E\big[d_G(W_t,e)\big]\ge ct^β, where \{W_t\}_{t=0}^\infty is the canonical simple random walk on the Cayley graph of G determined by S, starting at the identity element. This result is sharp when X=L_p, generalizes a theorem of Guentner and Kaminker and answers a question posed by Tessera. We also show that if α^*(G)\ge 1/2 then α^*(G\bwr \Z)\ge \frac{2α^*(G)}{2α^*(G)+1}. This improves the previous bound due to Stalder and ValetteWe deduce that if we write \Z_{(1)}= \Z and \Z_{(k+1)}\coloneqq \Z_{(k)}\bwr \Z then α^*(\Z_{(k)})=\frac{1}{2-2^{1-k}}, and use this result to answer a question posed by Tessera in on the relation between the Hilbert compression exponent and the isoperimetric profile of the balls in G. We also show that the cyclic lamplighter groups C_2\bwr C_n embed into L_1 with uniformly bounded distortion, answering a question posed by Lee, Naor and Peres. Finally, we use these results to show that edge Markov type need not imply Enflo type.

math.MG↗
arXiv2007

The wreath product of Z with Z has Hilbert compression exponent 2/3

Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all $α\ge 0$ such that there exists a Lipschitz mapping $f:G\to L_2$ and a constant $c>0$ such that for all $x,y\in G$ we have $\|f(x)-f(y)\|_2\ge cd(x,y)^α.$ In \cite{AGS06} it was shown that the Hilbert compression exponent of the wreath product $\Z\bwr \Z $ is at most $\frac34$, and in \cite{NP07} was proved that this exponent is at least $\frac23$. Here we show that $\frac23$ is the correct value. Our proof is based on an application of K. Ball's notion of Markov type.

math.MG↗
arXiv2007

On metric Ramsey-type phenomena

The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of largest cardinality which can be embedded with a given distortion in Hilbert space. We provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion. Our main theorem states that for any epsilon>0, every n point metric space contains a subset of size at least n^{1-ε} which is embeddable in Hilbert space with O(\frac{\log(1/ε)}ε) distortion. The bound on the distortion is tight up to the log(1/ε) factor. We further include a comprehensive study of various other aspects of this problem.

math.MG↗
arXiv2007

Trees and Markov convexity

We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant called Markov convexity, and show how it can be used to compute the Euclidean distortion of any metric tree up to universal factors.

math.MG↗
arXiv2007

The Euclidean distortion of the lamplighter group

We show that the cyclic lamplighter group $C_2 \bwr C_n$ embeds into Hilbert space with distortion ${\rm O}(\sqrt{\log n})$. This matches the lower bound proved by Lee, Naor and Peres in \cite{LeeNaoPer}, answering a question posed in that paper. Thus the Euclidean distortion of $C_2 \bwr C_n$ is $Θ(\sqrt{\log n})$. Our embedding is constructed explicitly in terms of the irreducible representations of the group. Since the optimal Euclidean embedding of a finite group can always be chosen to be equivariant, as shown by Aharoni, Maurey and Mityagin \cite{AhaMauMit} and by Gromov (see \cite{deCTesVal}), such representation-theoretic considerations suggest a general tool for obtaining upper and lower bounds on Euclidean embeddings of finite groups.

math.MG↗
arXiv2006

Ramsey partitions and proximity data structures

This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (a.k.a. the non-linear version of the isomorphic Dvoretzky theorem, as introduced by Bourgain, Figiel, and Milman). We then proceed to construct optimal Ramsey partitions, and use them to show that for every e\in (0,1), any n-point metric space has a subset of size n^{1-e} which embeds into Hilbert space with distortion O(1/e). This result is best possible and improves part of the metric Ramsey theorem of Bartal, Linial, Mendel and Naor, in addition to considerably simplifying its proof. We use our new Ramsey partitions to design the best known approximate distance oracles when the distortion is large, closing a gap left open by Thorup and Zwick. Namely, we show that for any $n$ point metric space X, and k>1, there exists an O(k)-approximate distance oracle whose storage requirement is O(n^{1+1/k}), and whose query time is a universal constant. We also discuss applications of Ramsey partitions to various other geometric data structure problems, such as the design of efficient data structures for approximate ranking.

cs.DS↗
arXiv2005

Lower bounds on Locality Sensitive Hashing

Given a metric space $(X,d_X)$, $c\ge 1$, $r>0$, and $p,q\in [0,1]$, a distribution over mappings $\h:X\to \mathbb N$ is called a $(r,cr,p,q)$-sensitive hash family if any two points in $X$ at distance at most $r$ are mapped by $\h$ to the same value with probability at least $p$, and any two points at distance greater than $cr$ are mapped by $\h$ to the same value with probability at most $q$. This notion was introduced by Indyk and Motwani in 1998 as the basis for an efficient approximate nearest neighbor search algorithm, and has since been used extensively for this purpose. The performance of these algorithms is governed by the parameter $ρ=\frac{\log(1/p)}{\log(1/q)}$, and constructing hash families with small $ρ$ automatically yields improved nearest neighbor algorithms. Here we show that for $X=\ell_1$ it is impossible to achieve $ρ\le \frac{1}{2c}$. This almost matches the construction of Indyk and Motwani which achieves $ρ\le \frac{1}{c}$.

cs.CG↗
arXiv2005

Nonembeddability theorems via Fourier analysis

Various new nonembeddability results (mainly into $L_1$) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on $\{0,1\}^d$ has $L_1$ distortion $(\log d)^{\frac12-o(1)}$. We also give new lower bounds on the $L_1$ distortion of flat tori, quotients of the discrete hypercube under group actions, and the transportation cost (Earthmover) metric.

math.FA↗
arXiv2005

Planar Earthmover is not in $L_1$

We show that any $L_1$ embedding of the transportation cost (a.k.a. Earthmover) metric on probability measures supported on the grid $\{0,1,...,n\}^2\subseteq \R^2$ incurs distortion $Ω(\sqrt{\log n})$. We also use Fourier analytic techniques to construct a simple $L_1$ embedding of this space which has distortion $O(\log n)$.

cs.CG↗
arXiv2005

Measured descent: A new embedding method for finite metrics

We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Frechet embeddings for finite metrics, due to [Bourgain, 1985] and [Rao, 1999]. We prove that any n-point metric space (X,d) embeds in Hilbert space with distortion O(sqrt{alpha_X log n}), where alpha_X is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an O(sqrt{(log lambda_X) \log n}) distortion embedding, where λ_X is the doubling constant of X. Since λ_X\le n, this result recovers Bourgain's theorem, but when the metric X is, in a sense, ``low-dimensional,'' improved bounds are achieved. Our embeddings are volume-respecting for subsets of arbitrary size. One consequence is the existence of (k, O(log n)) volume-respecting embeddings for all 1 \leq k \leq n, which is the best possible, and answers positively a question posed by U. Feige. Our techniques are also used to answer positively a question of Y. Rabinovich, showing that any weighted n-point planar graph embeds in l_\infty^{O(log n)} with O(1) distortion. The O(log n) bound on the dimension is optimal, and improves upon the previously known bound of O((log n)^2).

cs.DS↗
arXiv2005

Euclidean distortion and the Sparsest Cut

We prove that every $n$-point metric space of negative type (and, in particular, every $n$-point subset of $L_1$) embeds into a Euclidean space with distortion $O(\sqrt{\log n} \cdot\log \log n)$, a result which is tight up to the iterated logarithm factor. As a consequence, we obtain the best known polynomial-time approximation algorithm for the Sparsest Cut problem with general demands. Namely, if the demand is supported on a subset of size $k$, we achieve an approximation ratio of $O(\sqrt{\log k}\cdot \log \log k)$.

math.MG↗