200,206 research outputs found
Optimal Selection of Traffic Sensors: an Information-Theoretic Framework
This paper presents an information-theoretic framework for the optimal selection of sensors across a traffic network. For the selection of sensors a set covering integer programming (IP) problem is developed. A measure of correlation between random variables, reflecting a variable of interest, is introduced as a “distance” metric to provide sufficient coverage and information accuracy. The ultimate goal is to select sensors that are most informative about unsensed locations. The Kullback-Leibler divergence (relative entropy) is used to measure the dissimilarity between probability mass functions corresponding to different solutions of the IP program. Efficient model selection is a trade-off between the Kullback-Leibler divergence and the optimal cost of the IP program. The proposed framework is applied to the problem of developing sparse-measurement traffic flow models with empirical inductive loop-detector data of one week from a central business district with about sixty sensors. Results demonstrate that the obtained sparse-measurement rival models are able to preserve the shape and main features of the full-measurement traffic flow models
Extropy: Complementary Dual of Entropy
This article provides a completion to theories of information based on
entropy, resolving a longstanding question in its axiomatization as proposed by
Shannon and pursued by Jaynes. We show that Shannon's entropy function has a
complementary dual function which we call "extropy." The entropy and the
extropy of a binary distribution are identical. However, the measure bifurcates
into a pair of distinct measures for any quantity that is not merely an event
indicator. As with entropy, the maximum extropy distribution is also the
uniform distribution, and both measures are invariant with respect to
permutations of their mass functions. However, they behave quite differently in
their assessments of the refinement of a distribution, the axiom which
concerned Shannon and Jaynes. Their duality is specified via the relationship
among the entropies and extropies of course and fine partitions. We also
analyze the extropy function for densities, showing that relative extropy
constitutes a dual to the Kullback-Leibler divergence, widely recognized as the
continuous entropy measure. These results are unified within the general
structure of Bregman divergences. In this context they identify half the
metric as the extropic dual to the entropic directed distance. We describe a
statistical application to the scoring of sequential forecast distributions
which provoked the discovery.Comment: Published at http://dx.doi.org/10.1214/14-STS430 in the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A Modica-Mortola approximation for branched transport
The M^\alpha energy which is usually minimized in branched transport problems
among singular 1-dimensional rectifiable vector measures with prescribed
divergence is approximated (and convergence is proved) by means of a sequence
of elliptic energies, defined on more regular vector fields. The procedure
recalls the Modica-Mortola one for approximating the perimeter, and the
double-well potential is replaced by a concave power
Scalable Unbalanced Optimal Transport using Generative Adversarial Networks
Generative adversarial networks (GANs) are an expressive class of neural
generative models with tremendous success in modeling high-dimensional
continuous measures. In this paper, we present a scalable method for unbalanced
optimal transport (OT) based on the generative-adversarial framework. We
formulate unbalanced OT as a problem of simultaneously learning a transport map
and a scaling factor that push a source measure to a target measure in a
cost-optimal manner. In addition, we propose an algorithm for solving this
problem based on stochastic alternating gradient updates, similar in practice
to GANs. We also provide theoretical justification for this formulation,
showing that it is closely related to an existing static formulation by Liero
et al. (2018), and perform numerical experiments demonstrating how this
methodology can be applied to population modeling
Pushing fillings in right-angled Artin groups
We construct "pushing maps" on the cube complexes that model right-angled
Artin groups (RAAGs) in order to study filling problems in certain subsets of
these cube complexes. We use radial pushing to obtain upper bounds on higher
divergence functions, finding that the k-dimensional divergence of a RAAG is
bounded by r^{2k+2}. These divergence functions, previously defined for
Hadamard manifolds to measure isoperimetric properties "at infinity," are
defined here as a family of quasi-isometry invariants of groups; thus, these
results give new information about the QI classification of RAAGs. By pushing
along the height gradient, we also show that the k-th order Dehn function of a
Bestvina-Brady group is bounded by V^{(2k+2)/k}. We construct a class of RAAGs
called "orthoplex groups" which show that each of these upper bounds is sharp.Comment: The result on the Dehn function at infinity in mapping class groups
has been moved to the note "Filling loops at infinity in the mapping class
group.
Scaling Algorithms for Unbalanced Transport Problems
This article introduces a new class of fast algorithms to approximate
variational problems involving unbalanced optimal transport. While classical
optimal transport considers only normalized probability distributions, it is
important for many applications to be able to compute some sort of relaxed
transportation between arbitrary positive measures. A generic class of such
"unbalanced" optimal transport problems has been recently proposed by several
authors. In this paper, we show how to extend the, now classical, entropic
regularization scheme to these unbalanced problems. This gives rise to fast,
highly parallelizable algorithms that operate by performing only diagonal
scaling (i.e. pointwise multiplications) of the transportation couplings. They
are generalizations of the celebrated Sinkhorn algorithm. We show how these
methods can be used to solve unbalanced transport, unbalanced gradient flows,
and to compute unbalanced barycenters. We showcase applications to 2-D shape
modification, color transfer, and growth models
Non-perturbative \lambda\Phi^4 in D=1+1: an example of the constructive quantum field theory approach in a schematic way
During the '70, several relativistic quantum field theory models in
and also in have been constructed in a non-perturbative way. That was
done in the so-called {\it constructive quantum field theory} approach, whose
main results have been obtained by a clever use of Euclidean functional
methods. Although in the construction of a single model there are several
technical steps, some of them involving long proofs, the constructive quantum
field theory approach contains conceptual insights about relativistic quantum
field theory that deserved to be known and which are accessible without
entering in technical details. The purpose of this note is to illustrate such
insights by providing an oversimplified schematic exposition of the simple case
of (with ) in . Because of the absence of
ultraviolet divergences in its perturbative version, this simple example
-although does not capture all the difficulties in the constructive quantum
field theory approach- allows to stress those difficulties inherent to the
non-perturbative definition. We have made an effort in order to avoid several
of the long technical intermediate steps without missing the main ideas and
making contact with the usual language of the perturbative approach.Comment: 63 pages. Typos correcte
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