94,814 research outputs found

    Constructive quantization: approximation by empirical measures

    Get PDF
    In this article, we study the approximation of a probability measure μ\mu on Rd\mathbb{R}^{d} by its empirical measure μ^N\hat{\mu}_{N} interpreted as a random quantization. As error criterion we consider an averaged pp-th moment Wasserstein metric. In the case where 2p<d2p<d, 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

    Full text link
    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

    Get PDF
    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

    Get PDF
    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 nn-means, the nnth 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

    Full text link
    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

    Get PDF
    In this paper, we use quantization to construct a nonparametric estimator of conditional quantiles of a scalar response YY given a d-dimensional vector of covariates XX. First we focus on the population level and show how optimal quantization of XX, which consists in discretizing XX by projecting it on an appropriate grid of NN points, allows to approximate conditional quantiles of YY given XX. We show that this is approximation is arbitrarily good as NN 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-NN 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)
    • …
    corecore