arXiv subjects

James R. Lee

Publications and source records attributed to James R. Lee.

49 records · Page 3Linked to original sources

arXiv2011

Dimension reduction for finite trees in L_1

We show that every n-point tree metric admits a (1+eps)-embedding into a C(eps) log n-dimensional L_1 space, for every eps > 0, where C(eps) = O((1/eps)^4 log(1/eps)). This matches the natural volume lower bound up to a factor depending only on eps. Previously, it was unknown whether even complete binary trees on n nodes could be embedded in O(log n) dimensions with O(1) distortion. For complete d-ary trees, our construction achieves C(eps) = O(1/eps^2).

math.MG↗

arXiv2011

Metric uniformization and spectral bounds for graphs

We present a method for proving upper bounds on the eigenvalues of the graph Laplacian. A main step involves choosing an appropriate "Riemannian" metric to uniformize the geometry of the graph. In many interesting cases, the existence of such a metric is shown by examining the combinatorics of special types of flows. This involves proving new inequalities on the crossing number of graphs. In particular, we use our method to show that for any positive integer k, the kth smallest eigenvalue of the Laplacian on an n-vertex, bounded-degree planar graph is O(k/n). This bound is asymptotically tight for every k, as it is easily seen to be achieved for square planar grids. We also extend this spectral result to graphs with bounded genus, and graphs which forbid fixed minors. Previously, such spectral upper bounds were only known for the case k=2.

math.MG↗

arXiv2011

On the optimality of gluing over scales

We show that for every $α> 0$, there exist $n$-point metric spaces (X,d) where every "scale" admits a Euclidean embedding with distortion at most $α$, but the whole space requires distortion at least $Ω(\sqrt{α\log n})$. This shows that the scale-gluing lemma [Lee, SODA 2005] is tight, and disproves a conjecture stated there. This matching upper bound was known to be tight at both endpoints, i.e. when $α= Θ(1)$ and $α= Θ(\log n)$, but nowhere in between. More specifically, we exhibit $n$-point spaces with doubling constant $λ$ requiring Euclidean distortion $Ω(\sqrt{\log λ\log n})$, which also shows that the technique of "measured descent" [Krauthgamer, et. al., Geometric and Functional Analysis] is optimal. We extend this to obtain a similar tight result for $L_p$ spaces with $p > 1$.

math.MG↗

arXiv2010

Randomly removing g handles at once

Indyk and Sidiropoulos (2007) proved that any orientable graph of genus $g$ can be probabilistically embedded into a graph of genus $g-1$ with constant distortion. Viewing a graph of genus $g$ as embedded on the surface of a sphere with $g$ handles attached, Indyk and Sidiropoulos' method gives an embedding into a distribution over planar graphs with distortion $2^{O(g)}$, by iteratively removing the handles. By removing all $g$ handles at once, we present a probabilistic embedding with distortion $O(g^2)$ for both orientable and non-orientable graphs. Our result is obtained by showing that the nimum-cut graph of Erickson and Har Peled (2004) has low dilation, and then randomly cutting this graph out of the surface using the Peeling Lemma of Lee and Sidiropoulos (2009).

cs.CG↗

arXiv2009

Eigenvectors of random graphs: Nodal domains

We initiate a systematic study of eigenvectors of random graphs. Whereas much is known about eigenvalues of graphs and how they reflect properties of the underlying graph, relatively little is known about the corresponding eigenvectors. Our main focus in this paper is on the nodal domains associated with the different eigenfunctions. In the analogous realm of Laplacians of Riemannian manifolds, nodal domains have been the subject of intensive research for well over a hundred years. Graphical nodal domains turn out to have interesting and unexpected properties. Our main theorem asserts that there is a constant c such that for almost every graph G, each eigenfunction of G has at most two large nodal domains, and in addition at most c exceptional vertices outside these primary domains. We also discuss variations of these questions and briefly report on some numerical experiments which, in particular, suggest that almost surely there are just two nodal domains and no exceptional vertices.

math.PR↗

arXiv2009

Coarse differentiation and multi-flows in planar graphs

We show that the multi-commodity max-flow/min-cut gap for series-parallel graphs can be as bad as 2, matching a recent upper bound Chakrabarti, Jaffe, Lee, and Vincent for this class, and resolving one side of a conjecture of Gupta, Newman, Rabinovich, and Sinclair. This also improves the largest known gap for planar graphs from 3/2 to 2, yielding the first lower bound that doesn't follow from elementary calculations. Our approach uses the {\em coarse differentiation} method of Eskin, Fisher, and Whyte in order to lower bound the distortion for embedding a particular family of shortest-path metrics into $L_1$.

math.MG↗

arXiv2009

Almost Euclidean subspaces of \ell_1^N via expander codes

We give an explicit (in particular, deterministic polynomial time) construction of subspaces X of R^N of dimension (1-o(1))N such that for every element x in X, |x|_1 and N^{1/2} |x|_2 are equivalent up to a factor of (log N)^{log log log N}. If we are allowed to use N^{o(1)} random bits, this factor can be improved to poly(log N). Our construction makes use of unbalanced bipartite graphs to impose local linear constraints on vectors in the subspace, and our analysis relies on expansion properties of the graph. This is inspired by similar constructions of error-correcting codes.

math.MG↗

arXiv2009

Eigenvalue multiplicity and volume growth

Let $G$ be a finite group with symmetric generating set $S$, and let $c = \max_{R > 0} |B(2R)|/|B(R)|$ be the doubling constant of the corresponding Cayley graph, where $B(R)$ denotes an $R$-ball in the word-metric with respect to $S$. We show that the multiplicity of the $k$th eigenvalue of the Laplacian on the Cayley graph of $G$ is bounded by a function of only $c$ and $k$. More specifically, the multiplicity is at most $\exp((\log c)(\log c + \log k))$. Similarly, if $X$ is a compact, $n$-dimensional Riemannian manifold with non-negative Ricci curvature, then the multiplicity of the $k$th eigenvalue of the Laplace-Beltrami operator on $X$ is at most $\exp(n^2 + n log k)$. The first result (for $k=2$) yields the following group-theoretic application. There exists a normal subgroup $N$ of $G$, with $[G : N] \leq α(c)$, and such that $N$ admits a homomorphism onto the cyclic group $Z_M$, where $M \geq |G|^{δ(c)}$ and $α(c), δ(c) > 0$ are explicit functions depending only on $c$. This is the finitary analog of a theorem of Gromov which states that every infinite group of polynomial growth has a subgroup of finite index which admits a homomorphism onto the integers. This addresses a question of Trevisan, and is proved by scaling down Kleiner's proof of Gromov's theorem. In particular, we replace the space of harmonic functions of fixed polynomial growth by the second eigenspace of the Laplacian on the Cayley graph of $G$.

math.MG↗

arXiv2008

Eigenvalue bounds, spectral partitioning, and metrical deformations via flows

We present a new method for upper bounding the second eigenvalue of the Laplacian of graphs. Our approach uses multi-commodity flows to deform the geometry of the graph; we embed the resulting metric into Euclidean space to recover a bound on the Rayleigh quotient. Using this, we show that every $n$-vertex graph of genus $g$ and maximum degree $d$ satisfies $λ_2(G) = O((g+1)^3 d/n)$. This recovers the $O(d/n)$ bound of Spielman and Teng for planar graphs, and compares to Kelner's bound of $O((g+1) poly(d)/n)$, but our proof does not make use of conformal mappings or circle packings. We are thus able to extend this to resolve positively a conjecture of Spielman and Teng, by proving that $λ_2(G) = O(d h^6 \log h/n)$ whenever $G$ is $K_h$-minor free. This shows, in particular, that spectral partitioning can be used to recover $O(\sqrt{n})$-sized separators in bounded degree graphs that exclude a fixed minor. We extend this further by obtaining nearly optimal bounds on $λ_2$ for graphs which exclude small-depth minors in the sense of Plotkin, Rao, and Smith. Consequently, we show that spectral algorithms find small separators in a general class of geometric graphs. Moreover, while the standard "sweep" algorithm applied to the second eigenvector may fail to find good quotient cuts in graphs of unbounded degree, our approach produces a vector that works for arbitrary graphs. This yields an alternate proof of the result of Alon, Seymour, and Thomas that every excluded-minor family of graphs has $O(\sqrt{n})$-node balanced separators.

cs.DS↗

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↗

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↗

arXiv2004

Metric structures in L_1: Dimension, snowflakes, and average distortion

We study the metric properties of finite subsets of L_1. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation algorithms. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L_1.

math.MG↗