94,814 research outputs found
Constructive quantization: approximation by empirical measures
In this article, we study the approximation of a probability measure on
by its empirical measure interpreted as a
random quantization. As error criterion we consider an averaged -th moment
Wasserstein metric. In the case where , we establish refined upper and
lower bounds for the error, a high-resolution formula. Moreover, we provide a
universal estimate based on moments, a so-called Pierce type estimate. In
particular, we show that quantization by empirical measures is of optimal order
under weak assumptions.Comment: 22 page
Non-retracing orbits in Andreev billiards
The validity of the retracing approximation in the semiclassical quantization
of Andreev billiards is investigated. The exact energy spectrum and the
eigenstates of normal-conducting, ballistic quantum dots in contact with a
superconductor are calculated by solving the Bogoliubov-de Gennes equation and
compared with the semiclassical Bohr-Sommerfeld quantization for periodic
orbits which result from Andreev reflections. We find deviations that are due
to the assumption of exact retracing electron-hole orbits rather than the
semiclassical approximation, as a concurrently performed
Einstein-Brillouin-Keller quantization demonstrates. We identify three
different mechanisms producing non-retracing orbits which are directly
identified through differences between electron and hole wave functions.Comment: 9 pages, 12 figures, Phys. Rev. B (in print), high resolution images
available upon reques
SWKB Quantization Rules for Bound States in Quantum Wells
In a recent paper by Gomes and Adhikari (J.Phys B30 5987(1997)) a matrix
formulation of the Bohr-Sommerfield quantization rule has been applied to the
study of bound states in one dimension quantum wells. Here we study these
potentials in the frame work of supersymmetric WKB (SWKB) quantization
approximation and find that SWKB quantization rule is superior to the modified
Bohr-Sommerfield or WKB rules as it exactly reproduces the eigenenergies.Comment: 8 page
An overview of the quantization for mixed distributions
The basic goal of quantization for probability distribution is to reduce the
number of values, which is typically uncountable, describing a probability
distribution to some finite set and thus approximation of a continuous
probability distribution by a discrete distribution. Mixed distributions are an
exciting new area for optimal quantization. In this paper, we have determined
the optimal sets of -means, the th quantization error, and the
quantization dimensions of different mixed distributions. Besides, we have
discussed whether the quantization coefficients for the mixed distributions
exist. The results in this paper will give a motivation and insight into more
general problems in quantization of mixed distributions.Comment: arXiv admin note: text overlap with arXiv:1701.0416
Optimal Quantization of TV White Space Regions for a Broadcast Based Geolocation Database
In the current paradigm, TV white space databases communicate the available
channels over a reliable Internet connection to the secondary devices. For
places where an Internet connection is not available, such as in developing
countries, a broadcast based geolocation database can be considered. This
geolocation database will broadcast the TV white space (or the primary services
protection regions) on rate-constrained digital channel.
In this work, the quantization or digital representation of protection
regions is considered for rate-constrained broadcast geolocation database.
Protection regions should not be declared as white space regions due to the
quantization error. In this work, circular and basis based approximations are
presented for quantizing the protection regions. In circular approximation,
quantization design algorithms are presented to protect the primary from
quantization error while minimizing the white space area declared as protected
region. An efficient quantizer design algorithm is presented in this case. For
basis based approximations, an efficient method to represent the protection
regions by an `envelope' is developed. By design this envelope is a sparse
approximation, i.e., it has lesser number of non-zero coefficients in the basis
when compared to the original protection region. The approximation methods
presented in this work are tested using three experimental data-sets.Comment: 8 pages, 12 figures, submitted to IEEE DySPAN (Technology) 201
Conditional quantile estimation through optimal quantization
In this paper, we use quantization to construct a nonparametric estimator of
conditional quantiles of a scalar response given a d-dimensional vector of
covariates . First we focus on the population level and show how optimal
quantization of , which consists in discretizing by projecting it on an
appropriate grid of points, allows to approximate conditional quantiles of
given . We show that this is approximation is arbitrarily good as
goes to infinity and provide a rate of convergence for the approximation error.
Then we turn to the sample case and define an estimator of conditional
quantiles based on quantization ideas. We prove that this estimator is
consistent for its fixed- population counterpart. The results are
illustrated on a numerical example. Dominance of our estimators over local
constant/linear ones and nearest neighbor ones is demonstrated through
extensive simulations in the companion paper Charlier et al.(2014b)
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