7,754 research outputs found
Fast Construction of Nets in Low Dimensional Metrics, and Their Applications
We present a near linear time algorithm for constructing hierarchical nets in
finite metric spaces with constant doubling dimension. This data-structure is
then applied to obtain improved algorithms for the following problems:
Approximate nearest neighbor search, well-separated pair decomposition, compact
representation scheme, doubling measure, and computation of the (approximate)
Lipschitz constant of a function. In all cases, the running (preprocessing)
time is near-linear and the space being used is linear.Comment: 41 pages. Extensive clean-up of minor English error
Non-Euclidean geometry in nature
I describe the manifestation of the non-Euclidean geometry in the behavior of
collective observables of some complex physical systems. Specifically, I
consider the formation of equilibrium shapes of plants and statistics of sparse
random graphs. For these systems I discuss the following interlinked questions:
(i) the optimal embedding of plants leaves in the three-dimensional space, (ii)
the spectral statistics of sparse random matrix ensembles.Comment: 52 pages, 21 figures, last section is rewritten, a reference to
chaotic Hamiltonian systems is adde
Metric Cotype
We introduce the notion of metric cotype, a property of metric
spaces related to a property of normed spaces, called Rademacher
cotype. Apart from settling a long standing open problem in metric
geometry, this property is used to prove the following dichotomy: A
family of metric spaces F is either almost universal (i.e., contains
any finite metric space with any distortion > 1), or there exists
α > 0, and arbitrarily large n-point metrics whose distortion when
embedded in any member of F is at least Ω((log n)^α). The same
property is also used to prove strong non-embeddability theorems
of L_q into L_p, when q > max{2,p}. Finally we use metric cotype
to obtain a new type of isoperimetric inequality on the discrete
torus
Empirical geodesic graphs and CAT(k) metrics for data analysis
A methodology is developed for data analysis based on empirically constructed
geodesic metric spaces. For a probability distribution, the length along a path
between two points can be defined as the amount of probability mass accumulated
along the path. The geodesic, then, is the shortest such path and defines a
geodesic metric. Such metrics are transformed in a number of ways to produce
parametrised families of geodesic metric spaces, empirical versions of which
allow computation of intrinsic means and associated measures of dispersion.
These reveal properties of the data, based on geometry, such as those that are
difficult to see from the raw Euclidean distances. Examples of application
include clustering and classification. For certain parameter ranges, the spaces
become CAT(0) spaces and the intrinsic means are unique. In one case, a minimal
spanning tree of a graph based on the data becomes CAT(0). In another, a
so-called "metric cone" construction allows extension to CAT() spaces. It is
shown how to empirically tune the parameters of the metrics, making it possible
to apply them to a number of real cases.Comment: Statistics and Computing, 201
Approximate Nearest Neighbor Search for Low Dimensional Queries
We study the Approximate Nearest Neighbor problem for metric spaces where the
query points are constrained to lie on a subspace of low doubling dimension,
while the data is high-dimensional. We show that this problem can be solved
efficiently despite the high dimensionality of the data.Comment: 25 page
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