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  • Expected dispersion of uniformly distributed points
    J. Complex. (IF 0.888) Pub Date : 2020-04-02
    Aicke Hinrichs; David Krieg; Robert J. Kunsch; Daniel Rudolf

    The dispersion of a point set in [0,1]d is the volume of the largest axis parallel box inside the unit cube that does not intersect with the point set. We study the expected dispersion with respect to a random set of n points determined by an i.i.d. sequence of uniformly distributed random variables. Depending on the number of points n and the dimension d we provide an upper and lower bound of the

    更新日期:2020-04-03
  • On the power of standard information for L2-approximation in the average case setting
    J. Complex. (IF 0.888) Pub Date : 2020-03-13
    Guiqiao Xu

    This paper studies the equivalence of tractability for classes of Λstd of function evaluations and Λall of all continuous linear functionals for L2-approximation in the average-case setting. For the normalized error criterion, we show that the power of Λstd is the same as that of Λall for all notions of exponential tractability and algebraic tractability, in the sense that there exist algorithms using

    更新日期:2020-03-16
  • On the fixed volume discrepancy of the Fibonacci sets in the integral norms
    J. Complex. (IF 0.888) Pub Date : 2020-03-06
    V.N. Temlyakov; M. Ullrich

    This paper is devoted to the study of a discrepancy-type characteristic – the fixed volume discrepancy – of the Fibonacci point set in the unit square. It was observed recently that this new characteristic allows us to obtain optimal rate of dispersion from numerical integration results. This observation motivates us to thoroughly study this new version of discrepancy, which seems to be interesting

    更新日期:2020-03-06
  • The Diamond ensemble: A constructive set of spherical points with small logarithmic energy
    J. Complex. (IF 0.888) Pub Date : 2020-03-05
    Carlos Beltrán; Ujué Etayo

    We define a family of random sets of points, the Diamond ensemble, on the sphere S2 depending on several parameters. Its most important property is that, for some of these parameters, the asymptotic expected value of the logarithmic energy of the points can be computed rigorously and shown to attain very small values, quite close to the conjectured minimal value.

    更新日期:2020-03-05
  • Exponential convergence of adaptive quarklet approximation
    J. Complex. (IF 0.888) Pub Date : 2020-02-28
    Stephan Dahlke; Thorsten Raasch; Alexander Sieber

    This paper is concerned with approximation properties of polynomially enriched wavelet systems, so-called quarklet frames. We show that certain model singularities that arise in elliptic boundary value problems on polygonal domains can be approximated from the span of such quarklet systems at inverse-exponential rates. In order to realize these, we combine spatial refinement in the vicinity of the

    更新日期:2020-02-28
  • Iterative hard thresholding for compressed data separation
    J. Complex. (IF 0.888) Pub Date : 2020-02-28
    Song Li; Junhong Lin; Dekai Liu; Wenchang Sun

    We study the problem of reconstructing signals’ distinct subcomponents, which are approximately sparse in morphologically different dictionaries, from a small number of linear measurements. We propose an iterative hard thresholding algorithm adapted to dictionaries. We show that under the usual assumptions that the measurement system satisfies a restricted isometry property (adapted to a composed dictionary)

    更新日期:2020-02-28
  • Tractability properties of the discrepancy in Orlicz norms
    J. Complex. (IF 0.888) Pub Date : 2020-02-06
    Josef Dick; Aicke Hinrichs; Friedrich Pillichshammer; Joscha Prochno

    We show that the minimal discrepancy of a point set in the d-dimensional unit cube with respect to Orlicz norms can exhibit both polynomial and weak tractability. In particular, we show that the ψα-norms of exponential Orlicz spaces are polynomially tractable.

    更新日期:2020-02-07
  • Gauss-Newton methods with approximate projections for solving constrained nonlinear least squares problems
    J. Complex. (IF 0.888) Pub Date : 2020-01-21
    Max L.N. Gonçalves; Tiago C. Menezes

    This paper is concerned with algorithms for solving constrained nonlinear least squares problems. We first propose a local Gauss-Newton method with approximate projections for solving the aforementioned problems and study, by using a general majorant condition, its convergence results, including results on its rate. By combining the latter method and a nonmonotone line search strategy, we then propose

    更新日期:2020-01-22
  • Algorithms and complexity for functions on general domains
    J. Complex. (IF 0.888) Pub Date : 2020-01-21
    Erich Novak

    Error bounds and complexity bounds in numerical analysis and information-based complexity are often proved for functions that are defined on very simple domains, such as a cube, a torus, or a sphere. We study optimal error bounds for the approximation or integration of functions defined on Dd⊂Rd and only assume that Dd is a bounded Lipschitz domain. Some results are even more general. We study three

    更新日期:2020-01-22
  • Fast multivariate multi-point evaluation revisited
    J. Complex. (IF 0.888) Pub Date : 2019-04-22
    Joris van der Hoeven; Grégoire Lecerf

    In 2008, Kedlaya and Umans designed the first multivariate multi-point evaluation algorithm over finite fields with an asymptotic complexity that can be made arbitrarily close to linear. However, it remains a major challenge to make their algorithm efficient for practical input sizes. In this paper, we revisit and improve their algorithm, while keeping this ultimate goal in mind. In addition we sharpen

    更新日期:2020-01-04
  • Estimates of efficiency for two methods of stable numerical summation of smooth functions
    J. Complex. (IF 0.888) Pub Date : 2019-07-25
    S.G. Solodky; S.A. Stasyuk

    We consider a classical ill-posed problem of reconstruction of continuous functions from their noisy Fourier coefficients. We study the case of functions of two variables that has been much less investigated. The smoothness of reconstructed functions is measured in terms of the Sobolev classes as well as the classes of functions with dominated mixed derivatives. We investigate two summation methods

    更新日期:2020-01-04
  • On the complexity of extending the convergence region for Traub’s method
    J. Complex. (IF 0.888) Pub Date : 2019-07-29
    Ioannis K. Argyros; Santhosh George

    The convergence region of Traub’s method for solving equations is small in general. This fact limits its applicability. We locate a more precise region containing the Traub iterations leading to at least as tight Lipschitz constants as before. Our convergence analysis is finer, and obtained without additional conditions. The new theoretical results are tested on numerical examples that illustrate their

    更新日期:2020-01-04
  • Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs
    J. Complex. (IF 0.888) Pub Date : 2019-08-12
    Charles-Edouard Bréhier

    This article investigates the role of the regularity of the test function when considering the weak error for standard spatial and temporal discretizations of SPDEs of the form dX(t)=AX(t)dt+dW(t), driven by space–time white noise. In previous results, test functions are assumed (at least) of class C2 with bounded derivatives, and the weak order is twice the strong order. We prove that to quantify

    更新日期:2020-01-04
  • Optimal learning rates for distribution regression
    J. Complex. (IF 0.888) Pub Date : 2019-08-20
    Zhiying Fang; Zheng-Chu Guo; Ding-Xuan Zhou

    We study a learning algorithm for distribution regression with regularized least squares. This algorithm, which contains two stages of sampling, aims at regressing from distributions to real valued outputs. The first stage sample consists of distributions and the second stage sample is obtained from these distributions. To extract information from samples, we embed distributions to a reproducing kernel

    更新日期:2020-01-04
  • Absolute value information for IBC problems
    J. Complex. (IF 0.888) Pub Date : 2019-08-24
    Leszek Plaskota; Paweł Siedlecki; Henryk Woźniakowski

    Two classes of information have been mainly considered in Information-Based Complexity (IBC) for approximate solutions of continuous problems. The first class is Λall and consists of all linear functionals, whereas the second class is Λstd and consists of only function evaluations. A different class of information has been studied in the context of phase retrieval, where it is assumed that only absolute

    更新日期:2020-01-04
  • Stolarsky’s invariance principle for projective spaces
    J. Complex. (IF 0.888) Pub Date : 2019-09-05
    M.M. Skriganov

    We show that Stolarsky’s invariance principle, known for point distributions on the Euclidean spheres, can be extended to the real, complex, and quaternionic projective spaces and the octonionic projective plane.

    更新日期:2020-01-04
  • On the optimality of the trigonometric system
    J. Complex. (IF 0.888) Pub Date : 2019-09-07
    F. Jarad; A. Kushpel; K. Taş

    We study a new phenomenon of the behaviour of widths with respect to the optimality of trigonometric system. It is shown that the trigonometric system is optimal in the sense of Kolmogorov widths in the case of “super-high” and “super-small” smoothness but is not optimal in the intermediate cases. Bernstein’s widths behave differently when compared with Kolmogorov in the case of “super-small” smoothness

    更新日期:2020-01-04
  • Sampling schemes and recovery algorithms for functions of few coordinate variables
    J. Complex. (IF 0.888) Pub Date : 2019-12-27
    Simon Foucart

    When a multivariate function does not depend on all of its variables, it can be approximated from fewer point evaluations than otherwise required. This has been previously quantified e.g. in the case where the target function is Lipschitz. This note examines the same problem under other assumptions on the target function. If it is linear or quadratic, then connections to compressive sensing are exploited

    更新日期:2020-01-04
  • A note on the complexity of a phaseless polynomial interpolation
    J. Complex. (IF 0.888) Pub Date : 2019-11-29
    Michał R. Przybyłek; Paweł Siedlecki

    In this paper we revisit the classical problem of polynomial interpolation, with a slight twist; namely, polynomial evaluations are available up to a group action of the unit circle on the complex plane. It turns out that this new setting allows for a phaseless recovery of a polynomial in a polynomial time.

    更新日期:2020-01-04
  • Optimal approximation order of piecewise constants on convex partitions
    J. Complex. (IF 0.888) Pub Date : 2019-11-06
    Oleg Davydov; Oleksandr Kozynenko; Dmytro Skorokhodov

    We prove that the error of the best nonlinear Lp-approximation by piecewise constants on convex partitions is O(N−2d+1), where N is the number of cells, for all functions in the Sobolev space Wq2(Ω) on a cube Ω⊂Rd, d⩾2, as soon as 2d+1+1p−1q⩾0. The approximation order O(N−2d+1) is achieved on a polyhedral partition obtained by anisotropic refinement of an adaptive dyadic partition. Further estimates

    更新日期:2020-01-04
  • Information based complexity for high dimensional sparse functions
    J. Complex. (IF 0.888) Pub Date : 2019-11-02
    Cuize Han; Ming Yuan

    We investigate optimal algorithms for optimizing and approximating a general high dimensional smooth and sparse function from the perspective of information based complexity. Our algorithms and analyses reveal several interesting characteristics for these tasks. In particular, somewhat surprisingly, we show that the optimal sample complexity for optimization or high precision approximation is independent

    更新日期:2020-01-04
  • On local analysis
    J. Complex. (IF 0.888) Pub Date : 2019-10-28
    Felipe Cucker; Teresa Krick

    We extend to Gaussian distributions a result providing smoothed analysis estimates for condition numbers given as relativized distances to ill-posedness. We also introduce a notion of local analysis meant to capture the behavior of these condition numbers around a point.

    更新日期:2020-01-04
  • Counting points on hyperelliptic curves with explicit real multiplication in arbitrary genus
    J. Complex. (IF 0.888) Pub Date : 2019-10-21
    Simon Abelard

    We present a probabilistic Las Vegas algorithm for computing the local zeta function of a genus-g hyperelliptic curve defined over Fq with explicit real multiplication (RM) by an order Z[η] in a degree-g totally real number field. It is based on the approaches by Schoof and Pila in a more favourable case where we can split the ℓ-torsion into g kernels of endomorphisms, as introduced by Gaudry, Kohel

    更新日期:2020-01-04
  • A note on isotropic discrepancy and spectral test of lattice point sets
    J. Complex. (IF 0.888) Pub Date : 2019-10-18
    Friedrich Pillichshammer; Mathias Sonnleitner

    We show that the isotropic discrepancy of a lattice point set can be bounded from below and from above in terms of the spectral test of the corresponding integration lattice. From this result we deduce that the isotropic discrepancy of any N-element lattice point set in [0,1)d is at least of order N−1∕d. This order of magnitude is best possible for lattice point sets in dimension d.

    更新日期:2020-01-04
  • ε-superposition and truncation dimensions in average and probabilistic settings for ∞-variate linear problems
    J. Complex. (IF 0.888) Pub Date : 2019-10-08
    J. Dingess; G.W. Wasilkowski

    The paper deals with linear problems defined on γ-weighted Hilbert spaces of functions with infinitely many variables. The spaces are endowed with zero-mean Gaussian measures which allows to define and study ε-truncation and ε-superposition dimensions in the average case and probabilistic settings. Roughly speaking, these ε-dimensions quantify the smallest number k=k(ε) of variables that allow to approximate

    更新日期:2020-01-04
  • Lower error bounds for the stochastic gradient descent optimization algorithm: Sharp convergence rates for slowly and fast decaying learning rates
    J. Complex. (IF 0.888) Pub Date : 2019-09-27
    Arnulf Jentzen; Philippe von Wurstemberger

    The stochastic gradient descent (SGD) optimization algorithm is one of the central tools used to approximate solutions of stochastic optimization problems arising in machine learning and, in particular, deep learning applications. It is therefore important to analyze the convergence behavior of SGD. In this article we consider a simple quadratic stochastic optimization problem and establish for every

    更新日期:2020-01-04
  • Fast orthogonal transforms and generation of Brownian paths.
    J. Complex. (IF 0.888) Pub Date : 2013-03-09
    Gunther Leobacher

    We present a number of fast constructions of discrete Brownian paths that can be used as alternatives to principal component analysis and Brownian bridge for stratified Monte Carlo and quasi-Monte Carlo. By fast we mean that a path of length [Formula: see text] can be generated in [Formula: see text] floating point operations. We highlight some of the connections between the different constructions

    更新日期:2019-11-01
  • An Effective Algorithm for Generation of Factorial Designs with Generalized Minimum Aberration.
    J. Complex. (IF 0.888) Pub Date : 2007-01-01
    Kai-Tai Fang,Aijun Zhang,Runze Li

    Fractional factorial designs are popular and widely used for industrial experiments. Generalized minimum aberration is an important criterion recently proposed for both regular and non-regular designs. This paper provides a formal optimization treatment on optimal designs with generalized minimum aberration. New lower bounds and optimality results are developed for resolution-III designs. Based on

    更新日期:2019-11-01
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