4,085 research outputs found
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 \lambda_X is the doubling constant of X. Since \lambda_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).Comment: 17 pages. No figures. Appeared in FOCS '04. To appeaer in Geometric &
Functional Analysis. This version fixes a subtle error in Section 2.
Uniform Local Amenability
The main results of this paper show that various coarse (`large scale')
geometric properties are closely related. In particular, we show that property
A implies the operator norm localisation property, and thus that norms of
operators associated to a very large class of metric spaces can be effectively
estimated.
The main tool is a new property called uniform local amenability. This
property is easy to negate, which we use to study some `bad' spaces. We also
generalise and reprove a theorem of Nowak relating amenability and asymptotic
dimension in the quantitative setting
Convergence of resonances on thin branched quantum wave guides
We prove an abstract criterion stating resolvent convergence in the case of
operators acting in different Hilbert spaces. This result is then applied to
the case of Laplacians on a family X_\eps of branched quantum waveguides.
Combining it with an exterior complex scaling we show, in particular, that the
resonances on X_\eps approximate those of the Laplacian with ``free''
boundary conditions on , the skeleton graph of X_\eps.Comment: 48 pages, 1 figur
Large Unicellular maps in high genus
We study the geometry of a random unicellular map which is uniformly
distributed on the set of all unicellular maps whose genus size is proportional
to the number of edges of the map. We prove that the distance between two
uniformly selected vertices of such a map is of order and the diameter
is also of order with high probability. We further prove that the map
is locally planar with high probability. The main ingredient of the proofs is
an exploration procedure which uses a bijection due to Chapuy, Feray and Fusy.Comment: 43 pages, 6 figures, revised file taking into account referee's
comment
Consistency of Maximum Likelihood for Continuous-Space Network Models
Network analysis needs tools to infer distributions over graphs of arbitrary
size from a single graph. Assuming the distribution is generated by a
continuous latent space model which obeys certain natural symmetry and
smoothness properties, we establish three levels of consistency for
non-parametric maximum likelihood inference as the number of nodes grows: (i)
the estimated locations of all nodes converge in probability on their true
locations; (ii) the distribution over locations in the latent space converges
on the true distribution; and (iii) the distribution over graphs of arbitrary
size converges.Comment: 21 page
Classification of Minimal Separating Sets in Low Genus Surfaces
Consider a surface and let . If is not
connected, then we say \emph{separates} , and we refer to as a
\emph{separating set} of . If separates , and no proper subset of
separates , then we say is a \emph{minimal separating set} of . In
this paper we use methods of computational combinatorial topology to classify
the minimal separating sets of the orientable surfaces of genus and
. The classification for genus 0 and 1 was done in earlier work, using
methods of algebraic topology.Comment: 24 pages, 5 figures, 2 tables (11 pages
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