451,344 research outputs found
A New Weighted Metric: the Relative Metric I
The M-relative distance, denoted by \rho_M is a generalization of the
p-relative distance, which was introduced by Ren-Cang Li. We establish
necessary and sufficient conditions under which \rho_M is a metric. In two
special cases we derive complete characterizations of the metric. We also
present a way of extending the results to metrics sensitive to the domain in
which they are defined, thus finding some connections to previously studied
metrics.
An auxiliary result of independent interest is an inequality related to
Pittenger's inequality in Section 4.Comment: 23 page
Dissimilarity metric based on local neighboring information and genetic programming for data dissemination in vehicular ad hoc networks (VANETs)
This paper presents a novel dissimilarity metric based on local neighboring information
and a genetic programming approach for efficient data dissemination in Vehicular Ad Hoc Networks
(VANETs). The primary aim of the dissimilarity metric is to replace the Euclidean distance in
probabilistic data dissemination schemes, which use the relative Euclidean distance among vehicles
to determine the retransmission probability. The novel dissimilarity metric is obtained by applying a
metaheuristic genetic programming approach, which provides a formula that maximizes the Pearson
Correlation Coefficient between the novel dissimilarity metric and the Euclidean metric in several
representative VANET scenarios. Findings show that the obtained dissimilarity metric correlates with
the Euclidean distance up to 8.9% better than classical dissimilarity metrics. Moreover, the obtained
dissimilarity metric is evaluated when used in well-known data dissemination schemes, such as
p-persistence, polynomial and irresponsible algorithm. The obtained dissimilarity metric achieves
significant improvements in terms of reachability in comparison with the classical dissimilarity
metrics and the Euclidean metric-based schemes in the studied VANET urban scenarios
A Natural L(,p)-Metric for Spaces Composed of Probability Measures With P-Th Moment.
In the dissertation, it is seen that every probability measure with p-th moment on a complete, separable metric space can be viewed as a distribution of a metric space valued random variable. Between such random variables there exists an L(,p)-distance, and by finding the infimum of the L(,p)-distances between two types of random variables, it is possible to define a distance between distributions. It is seen that this distance can serve as a complete metric on the space of probability measures with p-th moment. The topology produced is shown to be equivalent to the topology of weak convergence of measures with convergence of moments. A closed embedding of this space into the space of finite measures with the weak convergence topology is produced and used to transfer Prokhorov\u27s theorem on tightness and relative compactness to the new setting. Other results, including a computational method for the case 1 (LESSTHEQ) p (LESSTHEQ) (INFIN) on the set of real numbers are also detailed and proven
Bayes and maximum likelihood for -Wasserstein deconvolution of Laplace mixtures
We consider the problem of recovering a distribution function on the real
line from observations additively contaminated with errors following the
standard Laplace distribution. Assuming that the latent distribution is
completely unknown leads to a nonparametric deconvolution problem. We begin by
studying the rates of convergence relative to the -norm and the Hellinger
metric for the direct problem of estimating the sampling density, which is a
mixture of Laplace densities with a possibly unbounded set of locations: the
rate of convergence for the Bayes' density estimator corresponding to a
Dirichlet process prior over the space of all mixing distributions on the real
line matches, up to a logarithmic factor, with the rate
for the maximum likelihood estimator. Then, appealing to an inversion
inequality translating the -norm and the Hellinger distance between
general kernel mixtures, with a kernel density having polynomially decaying
Fourier transform, into any -Wasserstein distance, , between the
corresponding mixing distributions, provided their Laplace transforms are
finite in some neighborhood of zero, we derive the rates of convergence in the
-Wasserstein metric for the Bayes' and maximum likelihood estimators of
the mixing distribution. Merging in the -Wasserstein distance between
Bayes and maximum likelihood follows as a by-product, along with an assessment
on the stochastic order of the discrepancy between the two estimation
procedures
On Deterministically Approximating Total Variation Distance
Total variation distance (TV distance) is an important measure for the
difference between two distributions. Recently, there has been progress in
approximating the TV distance between product distributions: a deterministic
algorithm for a restricted class of product distributions (Bhattacharyya,
Gayen, Meel, Myrisiotis, Pavan and Vinodchandran 2023) and a randomized
algorithm for general product distributions (Feng, Guo, Jerrum and Wang 2023).
We give a deterministic fully polynomial-time approximation algorithm (FPTAS)
for the TV distance between product distributions. Given two product
distributions and over , our algorithm
approximates their TV distance with relative error in time
.
Our algorithm is built around two key concepts: 1) The likelihood ratio as a
distribution, which captures sufficient information to compute the TV distance.
2) We introduce a metric between likelihood ratio distributions, called the
minimum total variation distance. Our algorithm computes a sparsified
likelihood ratio distribution that is close to the original one w.r.t. the new
metric. The approximated TV distance can be computed from the sparsified
likelihood ratio.
Our technique also implies deterministic FPTAS for the TV distance between
Markov chains
Barrlund's distance function and quasiconformal maps
Answering a question about triangle inequality suggested by R. Li, Barrlund [The p-relative distance is a metric. SIAM J Matrix Anal Appl. 1999;21:699.702] introduced a distance function which is a metric on a subdomain of We study this Barrlund metric and give sharp bounds for it in terms of other metrics of current interest. We also prove sharp distortion results for the Barrlund metric under quasiconformal maps
Geometric medians in reconciliation spaces
In evolutionary biology, it is common to study how various entities evolve
together, for example, how parasites coevolve with their host, or genes with
their species. Coevolution is commonly modelled by considering certain maps or
reconciliations from one evolutionary tree to another , all of which
induce the same map between the leaf-sets of and (corresponding
to present-day associations). Recently, there has been much interest in
studying spaces of reconciliations, which arise by defining some metric on
the set of all possible reconciliations between and .
In this paper, we study the following question: How do we compute a geometric
median for a given subset of relative to , i.e. an
element such that holds for all
? For a model where so-called host-switches or
transfers are not allowed, and for a commonly used metric called the
edit-distance, we show that although the cardinality of can be
super-exponential, it is still possible to compute a geometric median for a set
in in polynomial time. We expect that this result could
be useful for computing a summary or consensus for a set of reconciliations
(e.g. for a set of suboptimal reconciliations).Comment: 12 pages, 1 figur
Fuzzy dimensions and Planck's Uncertainty Principle for p-branes
The explicit form of the quantum propagator of a bosonic p-brane, previously
obtained by the authors in the quenched-minisuperspace approximation, suggests
the possibility of a novel, unified, description of p-branes with different
dimensionality. The background metric that emerges in this framework is a
quadratic form on a Clifford manifold. Substitution of the Lorentzian metric
with the Clifford line element has two far reaching consequences. On the one
hand, it changes the very structure of the spacetime fabric since the new
metric is built out of a Minimum Length below which it is impossible to resolve
the distance between two points; on the other hand, the introduction of the
Clifford line element extends the usual relativity of motion to the case of
Relative Dimensionalism of all p-branes that make up the spacetime manifold
near the Planck scale.Comment: 11 pages, LaTex, no figures; in print on Class.& Quantum Gra
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