5,617 research outputs found
Bregman Voronoi Diagrams: Properties, Algorithms and Applications
The Voronoi diagram of a finite set of objects is a fundamental geometric
structure that subdivides the embedding space into regions, each region
consisting of the points that are closer to a given object than to the others.
We may define many variants of Voronoi diagrams depending on the class of
objects, the distance functions and the embedding space. In this paper, we
investigate a framework for defining and building Voronoi diagrams for a broad
class of distance functions called Bregman divergences. Bregman divergences
include not only the traditional (squared) Euclidean distance but also various
divergence measures based on entropic functions. Accordingly, Bregman Voronoi
diagrams allow to define information-theoretic Voronoi diagrams in statistical
parametric spaces based on the relative entropy of distributions. We define
several types of Bregman diagrams, establish correspondences between those
diagrams (using the Legendre transformation), and show how to compute them
efficiently. We also introduce extensions of these diagrams, e.g. k-order and
k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set
of points and their connexion with Bregman Voronoi diagrams. We show that these
triangulations capture many of the properties of the celebrated Delaunay
triangulation. Finally, we give some applications of Bregman Voronoi diagrams
which are of interest in the context of computational geometry and machine
learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures
Geometric Set Cover and Hitting Sets for Polytopes in
Suppose we are given a finite set of points in and a collection of
polytopes that are all translates of the same polytope . We
consider two problems in this paper. The first is the set cover problem where
we want to select a minimal number of polytopes from the collection
such that their union covers all input points . The second
problem that we consider is finding a hitting set for the set of polytopes
, that is, we want to select a minimal number of points from the
input points such that every given polytope is hit by at least one point.
We give the first constant-factor approximation algorithms for both problems.
We achieve this by providing an epsilon-net for translates of a polytope in
of size \bigO(\frac{1{\epsilon)
Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and
present the first constant-ratio approximation for disks in the plane: Given a
set of disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . Our result is
the first constant-ratio approximation for a class of planar convex bodies of
arbitrary size and arbitrary intersections. In order to achieve a
-approximation, we reduce the traveling salesman problem with disks, up
to constant factors, to a minimum weight hitting set problem in a geometric
hypergraph. The connection between TSPN and hitting sets in geometric
hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure
Rho-estimators revisited: General theory and applications
Following Baraud, Birg\'e and Sart (2017), we pursue our attempt to design a
robust universal estimator of the joint ditribution of independent (but not
necessarily i.i.d.) observations for an Hellinger-type loss. Given such
observations with an unknown joint distribution and a dominated
model for , we build an estimator
based on and measure its risk by an
Hellinger-type distance. When does belong to the model, this risk
is bounded by some quantity which relies on the local complexity of the model
in a vicinity of . In most situations this bound corresponds to the
minimax risk over the model (up to a possible logarithmic factor). When
does not belong to the model, its risk involves an additional bias
term proportional to the distance between and ,
whatever the true distribution . From this point of view, this new
version of -estimators improves upon the previous one described in
Baraud, Birg\'e and Sart (2017) which required that be absolutely
continuous with respect to some known reference measure. Further additional
improvements have been brought as compared to the former construction. In
particular, it provides a very general treatment of the regression framework
with random design as well as a computationally tractable procedure for
aggregating estimators. We also give some conditions for the Maximum Likelihood
Estimator to be a -estimator. Finally, we consider the situation where
the Statistician has at disposal many different models and we build a penalized
version of the -estimator for model selection and adaptation purposes. In
the regression setting, this penalized estimator not only allows to estimate
the regression function but also the distribution of the errors.Comment: 73 page
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