119 research outputs found
Fat Polygonal Partitions with Applications to Visualization and Embeddings
Let be a rooted and weighted tree, where the weight of any node
is equal to the sum of the weights of its children. The popular Treemap
algorithm visualizes such a tree as a hierarchical partition of a square into
rectangles, where the area of the rectangle corresponding to any node in
is equal to the weight of that node. The aspect ratio of the
rectangles in such a rectangular partition necessarily depends on the weights
and can become arbitrarily high.
We introduce a new hierarchical partition scheme, called a polygonal
partition, which uses convex polygons rather than just rectangles. We present
two methods for constructing polygonal partitions, both having guarantees on
the worst-case aspect ratio of the constructed polygons; in particular, both
methods guarantee a bound on the aspect ratio that is independent of the
weights of the nodes.
We also consider rectangular partitions with slack, where the areas of the
rectangles may differ slightly from the weights of the corresponding nodes. We
show that this makes it possible to obtain partitions with constant aspect
ratio. This result generalizes to hyper-rectangular partitions in
. We use these partitions with slack for embedding ultrametrics
into -dimensional Euclidean space: we give a -approximation algorithm for embedding -point ultrametrics
into with minimum distortion, where denotes the spread
of the metric, i.e., the ratio between the largest and the smallest distance
between two points. The previously best-known approximation ratio for this
problem was polynomial in . This is the first algorithm for embedding a
non-trivial family of weighted-graph metrics into a space of constant dimension
that achieves polylogarithmic approximation ratio.Comment: 26 page
Multiscale likelihood analysis and complexity penalized estimation
We describe here a framework for a certain class of multiscale likelihood
factorizations wherein, in analogy to a wavelet decomposition of an L^2
function, a given likelihood function has an alternative representation as a
product of conditional densities reflecting information in both the data and
the parameter vector localized in position and scale. The framework is
developed as a set of sufficient conditions for the existence of such
factorizations, formulated in analogy to those underlying a standard
multiresolution analysis for wavelets, and hence can be viewed as a
multiresolution analysis for likelihoods. We then consider the use of these
factorizations in the task of nonparametric, complexity penalized likelihood
estimation. We study the risk properties of certain thresholding and
partitioning estimators, and demonstrate their adaptivity and near-optimality,
in a minimax sense over a broad range of function spaces, based on squared
Hellinger distance as a loss function. In particular, our results provide an
illustration of how properties of classical wavelet-based estimators can be
obtained in a single, unified framework that includes models for continuous,
count and categorical data types
Fat polygonal partitions with applications to visualization and embeddings
Let T be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node in T is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partition necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather than just rectangles. We present two methods for constructing polygonal partitions, both having guarantees on the worst-case aspect ratio of the constructed polygons; in particular, both methods guarantee a bound on the aspect ratio that is independent of the weights of the nodes. We also consider rectangular partitions with slack, where the areas of the rectangles may differ slightly from the weights of the corresponding nodes. We show that this makes it possible to obtain partitions with constant aspect ratio. This result generalizes to hyper-rectangular partitions in Rd. We use these partitions with slack for embedding ultrametrics into d-dimensional Euclidean space: we give a polylog(¿)-approximation algorithm for embedding n-point ultrametrics into Rd with minimum distortion, where ¿ denotes the spread of the metric. The previously best-known approximation ratio for this problem was polynomial in n. This is the first algorithm for embedding a non-trivial family of weighted-graph metrics into a space of constant dimension that achieves polylogarithmic approximation ratio
Higher Dimensional Coulomb Gases and Renormalized Energy Functionals
We consider a classical system of n charged particles in an external
confining potential, in any dimension d larger than 2. The particles interact
via pairwise repulsive Coulomb forces and the coupling parameter scales like
the inverse of n (mean-field scaling). By a suitable splitting of the
Hamiltonian, we extract the next to leading order term in the ground state
energy, beyond the mean-field limit. We show that this next order term, which
characterizes the fluctuations of the system, is governed by a new
"renormalized energy" functional providing a way to compute the total Coulomb
energy of a jellium (i.e. an infinite set of point charges screened by a
uniform neutralizing background), in any dimension. The renormalization that
cuts out the infinite part of the energy is achieved by smearing out the point
charges at a small scale, as in Onsager's lemma. We obtain consequences for the
statistical mechanics of the Coulomb gas: next to leading order asymptotic
expansion of the free energy or partition function, characterizations of the
Gibbs measures, estimates on the local charge fluctuations and factorization
estimates for reduced densities. This extends results of Sandier and Serfaty to
dimension higher than two by an alternative approach.Comment: Structure has slightly changed, details and corrections have been
added to some of the proof
Rectilinear minimum link paths in two and higher dimensions
The thesis discusses algorithms for the minimum link path problem, which is a well known geometric
path finding problem. The goal is to find a path that does the minimum number of turns amidst
obstacles in a continuous space. We focus on the most classical variant, the rectilinear minimum link
path problem, where the path and the obstacles are restricted to the directions of the coordinate
axes.
We study the rectilinear minimum link path problem in the plane and in the three-dimensional
space, as well as in higher dimensional domains. We present several new algorithms for solving
the problem in domains of varying dimension. For the planar case we develop a simple method
that has the optimal O(n log n) time complexity. For three-dimensional domains we present a new
algorithm with running time O(n^2 log^2 n), which is an improvement over the best previously known
result O(n^2.5 log n). The algorithm can also be generalized to higher dimensions, leading to an
O(n^(D-1) log^(D-1) n) time algorithm in D-dimensional domains.
We describe the new algorithms as well as the data structures used. The algorithms work by
maintaining a reachable region that is gradually expanded to form a shortest path map from the
starting point. The algorithms rely on several efficient data structures: the reachable region is
tracked by using a simple recursive space decomposition, and the region is expanded by a sweep
plane method that uses a multidimensional segment tree
Higher Dimensional Coulomb Gases and Renormalized Energy Functionals
Structure has slightly changed, details and corrections have been added to some of the proofs.We consider a classical system of n charged particles in an external confining potential, in any dimension d larger than 2. The particles interact via pairwise repulsive Coulomb forces and the coupling parameter scales like the inverse of n (mean-field scaling). By a suitable splitting of the Hamiltonian, we extract the next to leading order term in the ground state energy, beyond the mean-field limit. We show that this next order term, which characterizes the fluctuations of the system, is governed by a new "renormalized energy" functional providing a way to compute the total Coulomb energy of a jellium (i.e. an infinite set of point charges screened by a uniform neutralizing background), in any dimension. The renormalization that cuts out the infinite part of the energy is achieved by smearing out the point charges at a small scale, as in Onsager's lemma. We obtain consequences for the statistical mechanics of the Coulomb gas: next to leading order asymptotic expansion of the free energy or partition function, characterizations of the Gibbs measures, estimates on the local charge fluctuations and factorization estimates for reduced densities. This extends results of Sandier and Serfaty to dimension higher than two by an alternative approach
- …