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  • Multi-task Learning in vector-valued reproducing kernel Banach spaces with the ℓ1 norm
    J. Complex. (IF 1.338) Pub Date : 2020-09-03
    Rongrong Lin; Guohui Song; Haizhang Zhang

    Targeting at sparse multi-task learning, we consider regularization models with an ℓ1 penalty on the coefficients of kernel functions. In order to provide a kernel method for this model, we construct a class of vector-valued reproducing kernel Banach spaces with the ℓ1 norm. The notion of multi-task admissible kernels is proposed so that the constructed spaces could have desirable properties including

    更新日期:2020-09-03
  • A closer look at covering number bounds for Gaussian kernels
    J. Complex. (IF 1.338) Pub Date : 2020-07-24
    Ingo Steinwart; Simon Fischer

    We establish some new bounds on the log-covering numbers of (anisotropic) Gaussian reproducing kernel Hilbert spaces. Unlike previous results in this direction we focus on small explicit constants and their dependency on crucial parameters such as the kernel bandwidth and the size and dimension of the underlying space.

    更新日期:2020-07-24
  • Optimal subsampling for large-scale quantile regression
    J. Complex. (IF 1.338) Pub Date : 2020-07-21
    Mingyao Ai; Fei Wang; Jun Yu; Huiming Zhang

    To deal with massive data sets, subsampling is known as an effective method which can significantly reduce computational costs in estimating model parameters. In this article, an efficient subsampling method is developed for large-scale quantile regression via Poisson sampling framework, which can solve the memory constraint problem imposed by big data. Under some mild conditions, large sample properties

    更新日期:2020-07-21
  • The radii of sections of origin-symmetric convex bodies and their applications
    J. Complex. (IF 1.338) Pub Date : 2020-07-09
    Alexander Kushpel; Kenan Taş

    Let V and W be any convex and origin-symmetric bodies in Rn . Assume that for some A ∈LRn→Rn, detA≠0, V is contained in the ellipsoid A−1B2n, where B2n is the unit Euclidean ball. We give a lower bound for the W-radius of sections of A−1V in terms of the spectral radius of A∗A and the expectations of ‖⋅‖V and ‖⋅‖Wo with respect to Haar measure on Sn−1⊂Rn. It is shown that the respective expectations

    更新日期:2020-07-10
  • Instances of computational optimal recovery: Refined approximability models
    J. Complex. (IF 1.338) Pub Date : 2020-07-03
    Simon Foucart

    Models based on approximation capabilities have recently been studied in the context of Optimal Recovery. These models, however, are not compatible with overparametrization, since model- and data-consistent functions could then be unbounded. This drawback motivates the introduction of refined approximability models featuring an added boundedness condition. Thus, two new models are proposed in this

    更新日期:2020-07-03
  • Computing syzygies in finite dimension using fast linear algebra
    J. Complex. (IF 1.338) Pub Date : 2020-06-17
    Vincent Neiger, Éric Schost

    We consider the computation of syzygies of multivariate polynomials in a finite-dimensional setting: for a K[X1,…,Xr]-module M of finite dimension D as a K-vector space, and given elements f1,…,fm in M, the problem is to compute syzygies between the fi’s, that is, polynomials (p1,…,pm) in K[X1,…,Xr]m such that p1f1+⋯+pmfm=0 in M. Assuming that the multiplication matrices of the r variables with respect

    更新日期:2020-06-17
  • Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals
    J. Complex. (IF 1.338) Pub Date : 2020-06-11
    Peter Kritzer; Friedrich Pillichshammer; Henryk Woźniakowski

    We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an ε-approximation for the d-variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information

    更新日期:2020-06-11
  • Quasi-Monte Carlo and ε-truncation dimension in ANOVA spaces
    J. Complex. (IF 1.338) Pub Date : 2020-05-30
    G.W. Wasilkowski

    We explain the efficiency of quasi-Monte Carlo methods for high dimensional integration by proving that the worst case ε-truncation dimension of the error functional on γ-weighted ANOVA spaces is small for modest error demands.

    更新日期:2020-05-30
  • Fast computation of generic bivariate resultants
    J. Complex. (IF 1.338) Pub Date : 2020-05-30
    Joris van der Hoeven; Grégoire Lecerf

    We prove that the resultant of two “sufficiently generic” bivariate polynomials over a finite field can be computed in quasi-linear expected time, using a randomized algorithm of Las Vegas type. A similar complexity bound is proved for the computation of the lexicographical Gröbner basis for the ideal generated by the two polynomials.

    更新日期:2020-05-30
  • Directed evaluation
    J. Complex. (IF 1.338) Pub Date : 2020-05-26
    Joris van der Hoeven, Grégoire Lecerf

    Let K be a fixed effective field. The most straightforward approach to compute with an element in the algebraic closure of K is to compute modulo its minimal polynomial. The determination of a minimal polynomial from an arbitrary annihilator requires an algorithm for polynomial factorization over K. Unfortunately, such algorithms do not exist over generic effective fields. They do exist over fields

    更新日期:2020-05-26
  • Complexity of approximating Hölder classes from information with varying Gaussian noise
    J. Complex. (IF 1.338) Pub Date : 2020-05-21
    Paweł M. Morkisz, Leszek Plaskota

    We consider information ε-complexity of Lp-approximation (1≤p<∞) of functions f∈Cr([0,1]d) for which all the partial derivatives of order r are Hölder continuous with exponent ϱ∈(0,1]. Allowable approximations use nonadaptive or adaptive observations of f at finitely many points, and these observations are additionally contaminated by Gaussian noise. The cost of a single observation depends on the

    更新日期:2020-05-21
  • Strongly asymptotically optimal schemes for the strong approximation of stochastic differential equations with respect to the supremum error
    J. Complex. (IF 1.338) Pub Date : 2020-05-15
    Simon Hatzesberger

    Our subject of study is strong approximation of stochastic differential equations (SDEs) with respect to the supremum error criterion, and we seek approximations that are strongly asymptotically optimal in specific classes of approximations. We hereby focus on two principal types of classes, namely, the classes of approximations that are based only on the evaluation of the initial value and on at most

    更新日期:2020-05-15
  • Positive-Definite Functions, Exponential Sums and the Greedy Algorithm: a Curious Phenomenon
    J. Complex. (IF 1.338) Pub Date : 2020-04-23
    Louis Brown; Stefan Steinerberger

    We describe a curious dynamical system that results in sequences of real numbers in [0,1] with seemingly remarkable properties. Let the even function f:T→R satisfy f̂(k)≥c|k|−2 and define a sequence via xn=argminx∑k=1n−1f(x−xk). Such sequences (xn)n=1∞ seem to be astonishingly regularly distributed in various ways (satisfying favorable exponential sum estimates; every interval J⊂[0,1] contains ∼|J|n

    更新日期:2020-04-23
  • On the worst-case error of least squares algorithms for L2-approximation with high probability
    J. Complex. (IF 1.338) Pub Date : 2020-04-18
    Mario Ullrich

    It was recently shown by D. Krieg and M. Ullrich that, for L2-approximation of functions from a reproducing kernel Hilbert space, function values are almost as powerful as arbitrary linear information if the approximation numbers are square-summable. That is, en≲1kn∑j≥knaj2withkn≍nln(n),where en are the sampling numbers and ak are the approximation numbers. In particular, if (ak)∈ℓ2, then en and an

    更新日期:2020-04-18
  • Expected dispersion of uniformly distributed points
    J. Complex. (IF 1.338) 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 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 a lower bound of the expected

    更新日期:2020-04-02
  • On the power of standard information for L2-approximation in the average case setting
    J. Complex. (IF 1.338) 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:2020-03-13
  • On the fixed volume discrepancy of the Fibonacci sets in the integral norms
    J. Complex. (IF 1.338) 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 1.338) 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 1.338) 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 1.338) 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 1.338) 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:2020-02-06
  • Gauss–Newton methods with approximate projections for solving constrained nonlinear least squares problems
    J. Complex. (IF 1.338) 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:2020-01-21
  • Algorithms and complexity for functions on general domains
    J. Complex. (IF 1.338) 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:2020-01-21
  • Sampling schemes and recovery algorithms for functions of few coordinate variables
    J. Complex. (IF 1.338) 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:2019-12-27
  • A note on the complexity of a phaseless polynomial interpolation
    J. Complex. (IF 1.338) 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:2019-11-29
  • Optimal approximation order of piecewise constants on convex partitions
    J. Complex. (IF 1.338) 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:2019-11-06
  • Information based complexity for high dimensional sparse functions
    J. Complex. (IF 1.338) 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:2019-11-02
  • An Effective Algorithm for Generation of Factorial Designs with Generalized Minimum Aberration.
    J. Complex. (IF 1.338) 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
  • On local analysis
    J. Complex. (IF 1.338) 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:2019-10-28
  • Counting points on hyperelliptic curves with explicit real multiplication in arbitrary genus
    J. Complex. (IF 1.338) 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:2019-10-21
  • A note on isotropic discrepancy and spectral test of lattice point sets
    J. Complex. (IF 1.338) 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:2019-10-18
  • ε-superposition and truncation dimensions in average and probabilistic settings for ∞-variate linear problems
    J. Complex. (IF 1.338) 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:2019-10-08
  • Lower error bounds for the stochastic gradient descent optimization algorithm: Sharp convergence rates for slowly and fast decaying learning rates
    J. Complex. (IF 1.338) 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:2019-09-27
  • On the optimality of the trigonometric system
    J. Complex. (IF 1.338) 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:2019-09-07
  • Stolarsky’s invariance principle for projective spaces
    J. Complex. (IF 1.338) 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:2019-09-05
  • Absolute value information for IBC problems
    J. Complex. (IF 1.338) 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:2019-08-24
  • Optimal learning rates for distribution regression
    J. Complex. (IF 1.338) 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:2019-08-20
  • EC-(s,t)-weak tractability of multivariate linear problems in the average case setting
    J. Complex. (IF 1.338) Pub Date : 2019-08-13
    Anargyros Papageorgiou, Iasonas Petras, Guiqiao Xu, Dong Yanqi

    We study EC-(s,t)-weak tractability of multivariate linear problems in the average case setting. This paper extends earlier work in the worst case setting. The parameters s≥0 and t≥0 allow us to study the information complexity n(ε,d) of a d-variate problem with respect to different powers of lnε−1, corresponding to the bits of accuracy, and d. We consider the absolute and normalized error criteria

    更新日期:2019-08-13
  • Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs
    J. Complex. (IF 1.338) 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:2019-08-12
  • On the complexity of extending the convergence region for Traub’s method
    J. Complex. (IF 1.338) 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:2019-07-29
  • Estimates of efficiency for two methods of stable numerical summation of smooth functions
    J. Complex. (IF 1.338) 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:2019-07-25
  • A note about EC-(s,t)-weak tractability of multivariate approximation with analytic Korobov kernels
    J. Complex. (IF 1.338) Pub Date : 2019-07-02
    Heping Wang

    This paper is devoted to discussing multivariate approximation problems with analytic Korobov kernels in the worst and average case settings. We consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We investigate exponential convergence-(s,t)-weak tractability (EC-(s,t)-WT) under the absolute or normalized error criterion. We completely solve the problem

    更新日期:2019-07-02
  • EC-tractability of approximation problems in function spaces defined over products of simplices
    J. Complex. (IF 1.338) Pub Date : 2019-06-19
    Yongping Liu, Jie Zhang

    Many authors have studied exponentially-convergent tractability in the worst case setting. But almost all articles related to tractability are focused on the classes of some multivariate functions defined over the unit cubes [0,1]m or Rm. Note that simplex is also important in geometry and has many practical applications. In this paper we considered tractability of the approximation problems of function

    更新日期:2019-06-19
  • A nonlocal functional promoting low-discrepancy point sets
    J. Complex. (IF 1.338) Pub Date : 2019-06-11
    Stefan Steinerberger

    Let X=x1,…,xN⊂Td≅[0,1]d be a set of N points in the d−dimensional torus that we want to arrange as regularly possible. The purpose of this paper is to introduce the energy functional E(X)=∑1≤m,n≤Nm≠n∏k=1d(1−log2sinπ|xm,k−xn,k|) and to suggest that moving a set X into the direction −∇E(X) may have the effect of increasing regularity of the set in the sense of decreasing discrepancy. We numerically demonstrate

    更新日期:2019-06-11
  • Sampling discretization error of integral norms for function classes
    J. Complex. (IF 1.338) Pub Date : 2019-05-24
    V.N. Temlyakov

    The new ingredient of this paper is that we consider infinitely dimensional classes of functions and instead of the relative error setting, which was used in previous papers on norm discretization, we consider the absolute error setting. We demonstrate how known results from two areas of research – supervised learning theory and numerical integration – can be used in sampling discretization of the

    更新日期:2019-05-24
  • Extremal distributions of discrepancy functions
    J. Complex. (IF 1.338) Pub Date : 2019-05-23
    Ralph Kritzinger, Markus Passenbrunner

    The irregularities of a distribution of N points in the unit interval are often measured with various notions of discrepancy. The discrepancy function can be defined with respect to intervals of the form [0,t)⊂[0,1) or arbitrary subintervals of the unit interval. In the former case, it is a well known fact in discrepancy theory that the N-element point set in the unit interval with the lowest L2 or

    更新日期:2019-05-23
  • Embeddings for infinite-dimensional integration and L2-approximation with increasing smoothness
    J. Complex. (IF 1.338) Pub Date : 2019-05-16
    M. Gnewuch, M. Hefter, A. Hinrichs, K. Ritter, G.W. Wasilkowski

    We study integration and L2-approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh, Haar, and Sobolev spaces. For the proofs we derive embedding theorems between spaces of increasing smoothness and appropriate weighted function spaces of fixed smoothness

    更新日期:2019-05-16
  • Central limit theorems for multilevel Monte Carlo methods
    J. Complex. (IF 1.338) Pub Date : 2019-05-16
    Håkon Hoel, Sebastian Krumscheid

    In this work, we show that uniform integrability is not a necessary condition for central limit theorems (CLT) to hold for normalized multilevel Monte Carlo (MLMC) estimators and we provide near optimal weaker conditions under which the CLT is achieved. In particular, if the variance decay rate dominates the computational cost rate (i.e., β>γ), we prove that the CLT applies to the standard (variance

    更新日期:2019-05-16
  • Fast multivariate multi-point evaluation revisited
    J. Complex. (IF 1.338) 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:2019-04-22
  • Faster polynomial multiplication over finite fields using cyclotomic coefficient rings
    J. Complex. (IF 1.338) Pub Date : 2019-04-04
    David Harvey, Joris van der Hoeven

    We prove that for a fixed prime p, polynomials in Fp[X] of degree n may be multiplied in O(nlogn4log∗n) bit operations. Previously, the best known bound was O(nlogn8log∗n).

    更新日期:2019-04-04
  • Accelerated tower arithmetic
    J. Complex. (IF 1.338) Pub Date : 2019-03-18
    Joris van der Hoeven, Grégoire Lecerf

    Nowadays, asymptotically fast algorithms are widely used in computer algebra for computations in towers of algebraic field extensions of small height. Yet it is still unknown how to reach softly linear time for products and inversions in towers of arbitrary height. In this paper we design the first algorithm for general ground fields with a complexity exponent that can be made arbitrarily close to

    更新日期:2019-03-18
  • Normal numbers and nested perfect necklaces
    J. Complex. (IF 1.338) Pub Date : 2019-03-18
    Verónica Becher, Olivier Carton

    M. B. Levin used Sobol–Faure low discrepancy sequences with Pascal triangle matrices modulo 2 to construct, a real number x such that the first N terms of the sequence (2nxmod1)n≥1 have discrepancy O((logN)2∕N). This is the lowest discrepancy known for this kind of sequences. In this note we characterize Levin’s construction in terms of nested perfect necklaces, which are a variant of the classical

    更新日期:2019-03-18
  • Construction of highly redundant incoherent unit norm tight frames as a union of orthonormal bases
    J. Complex. (IF 1.338) Pub Date : 2019-03-15
    Pradip Sasmal, Phanindra Jampana, C.S. Sastry

    Unit norm tight frames are useful in providing stable reconstruction. Incoherence, restricted isometry property and statistical restricted isometry property of a frame provide performance guarantees of sparse signal recovery algorithms in compressed sensing. Equiangular tight frames, mutually unbiased bases and chirp matrices are well known incoherent unit norm tight frames that exhibit near optimal

    更新日期:2019-03-15
  • Generalized Kung–Traub method and its multi-step iteration in Banach spaces
    J. Complex. (IF 1.338) Pub Date : 2019-03-01
    Janak Raj Sharma, Sunil Kumar, Ioannis K. Argyros

    We generalize an optimal fourth order Kung–Traub method to Banach spaces and study its local convergence to approximate a locally-unique solution of a system of nonlinear equations. New analysis also provides radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions of higher order derivatives. Furthermore

    更新日期:2019-03-01
  • Approximation complexity of sums of random processes
    J. Complex. (IF 1.338) Pub Date : 2019-02-28
    A.A. Khartov, M. Zani

    We study approximation properties of additive random fields Yd(t),t∈[0,1]d, d∈N, which are sums of d uncorrelated zero-mean random processes with continuous covariance functions. The average case approximation complexity nYd(ε) is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate Yd, with relative 2-average error not exceeding a given threshold ε∈(0

    更新日期:2019-02-28
  • Quantitative bounds for concentration-of-measure inequalities and empirical regression: The independent case
    J. Complex. (IF 1.338) Pub Date : 2019-02-19
    David Barrera, Emmanuel Gobet

    This paper is devoted to the study of the deviation of the (random) average L2−error associated to the least-squares regressor over a family of functions Fn (with controlled complexity) obtained from n independent, but not necessarily identically distributed, samples of explanatory and response variables, from the minimal (deterministic) average L2−error associated to this family of functions, and

    更新日期:2019-02-19
  • Sharp estimates for approximation numbers of non-periodic Sobolev embeddings
    J. Complex. (IF 1.338) Pub Date : 2019-02-18
    Therese Mieth

    We investigate asymptotically sharp upper and lower bounds for the approximation numbers of the compact Sobolev embeddings W∘m(Ω)↪L2(Ω) and Wm(Ω)↪L2(Ω), defined on a bounded domain Ω⊂Rd, involving explicit constants depending on m and d. The key of proof is to relate the approximation problems to certain Dirichlet and Neumann eigenvalue problems.

    更新日期:2019-02-18
  • Random bit multilevel algorithms for stochastic differential equations
    J. Complex. (IF 1.338) Pub Date : 2019-01-30
    Michael B. Giles, Mario Hefter, Lukas Mayer, Klaus Ritter

    We study the approximation of expectations E(f(X)) for solutions X of SDEs and functionals f:C([0,1],Rr)→R by means of restricted Monte Carlo algorithms that may only use random bits instead of random numbers. We consider the worst case setting for functionals f from the Lipschitz class w.r.t. the supremum norm. We construct a random bit multilevel Euler algorithm and establish upper bounds for its

    更新日期:2019-01-30
  • The isotropic constant of random polytopes with vertices on convex surfaces
    J. Complex. (IF 1.338) Pub Date : 2019-01-29
    Joscha Prochno, Christoph Thäle, Nicola Turchi

    For an isotropic convex body K⊂Rn we consider the isotropic constant LKN of the symmetric random polytope KN generated by N independent random points which are distributed according to the cone probability measure on the boundary of K. We show that with overwhelming probability LKN≤Clog(2N∕n), where C∈(0,∞) is an absolute constant. If K is unconditional we argue that even LKN≤C with overwhelming probability

    更新日期:2019-01-29
  • Generalized adaptive versus nonadaptive recovery from noisy information
    J. Complex. (IF 1.338) Pub Date : 2019-01-06
    K. Yu. Osipenko

    The article is devoted to a generalized setting of adaptive recovery from noisy information. We extend some known results for this setting and obtain sufficient conditions under which adaption does not help for recovery of linear operators. For recovery of linear functionals we give necessary and sufficient conditions for the existence of an optimal nonadaptive linear method which gives the same error

    更新日期:2019-01-06
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