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1.
We consider the problem of approximately reconstructing a function f defined on the surface of the unit sphere in the Euclidean space ℝq +1 by using samples of f at scattered sites. A central role is played by the construction of a new operator for polynomial approximation, which is a uniformly bounded quasi‐projection in the de la Vallée Poussin style, i.e. it reproduces spherical polynomials up to a certain degree and has uniformly bounded Lp operator norm for 1 ≤ p ≤ ∞. Using certain positive quadrature rules for scattered sites due to Mhaskar, Narcowich and Ward, we discretize this operator obtaining a polynomial approximation of the target function which can be computed from scattered data and provides the same approximation degree of the best polynomial approximation. To establish the error estimates we use Marcinkiewicz–Zygmund inequalities, which we derive from our continuous approximating operator. We give concrete bounds for all constants in the Marcinkiewicz–Zygmund inequalities as well as in the error estimates. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
2.
PowerNet: Efficient Representations of Polynomials and Smooth Functions by Deep Neural Networks with Rectified Power Units 下载免费PDF全文
Deep neural network with rectified linear units (ReLU) is getting more and
more popular recently. However, the derivatives of the function represented by a ReLU
network are not continuous, which limit the usage of ReLU network to situations only
when smoothness is not required. In this paper, we construct deep neural networks
with rectified power units (RePU), which can give better approximations for smooth
functions. Optimal algorithms are proposed to explicitly build neural networks with
sparsely connected RePUs, which we call PowerNets, to represent polynomials with
no approximation error. For general smooth functions, we first project the function to
their polynomial approximations, then use the proposed algorithms to construct corresponding PowerNets. Thus, the error of best polynomial approximation provides an
upper bound of the best RePU network approximation error. For smooth functions in
higher dimensional Sobolev spaces, we use fast spectral transforms for tensor-product
grid and sparse grid discretization to get polynomial approximations. Our constructive algorithms show clearly a close connection between spectral methods and deep
neural networks: PowerNets with $n$ hidden layers can exactly represent polynomials
up to degree $s^n$, where $s$ is the power of RePUs. The proposed PowerNets have potential applications in the situations where high-accuracy is desired or smoothness is
required. 相似文献
3.
Yafang Gong 《分析论及其应用》2006,22(4)
Based on Bernstein's Theorem, Kalandia's Lemma describes the error estimate and the smoothness of the remainder under the second part of Holder norm when a Holder function is approximated by its best polynomial approximation. In this paper, Kalandia's Lemma is generalized to the cases that the best polynomial is replaced by one of its four kinds of Chebyshev polynomial expansions, the error estimates of the remainder are given out under Holder norm or the weighted Holder norms. 相似文献
4.
Jan Beirlant 《Journal of multivariate analysis》2004,89(1):97-118
We discuss the estimation of the tail index of a heavy-tailed distribution when covariate information is available. The approach followed here is based on the technique of local polynomial maximum likelihood estimation. The generalized Pareto distribution is fitted locally to exceedances over a high specified threshold. The method provides nonparametric estimates of the parameter functions and their derivatives up to the degree of the chosen polynomial. Consistency and asymptotic normality of the proposed estimators will be proven under suitable regularity conditions. This approach is motivated by the fact that in some applications the threshold should be allowed to change with the covariates due to significant effects on scale and location of the conditional distributions. Using the asymptotic results we are able to derive an expression for the asymptotic mean squared error, which can be used to guide the selection of the bandwidth and the threshold. The applicability of the method will be demonstrated with a few practical examples. 相似文献
5.
In the context of adaptive nonparametric curve estimation a common assumption is that a function (signal) to estimate belongs to a nested family of functional classes. These classes are often parametrized by a quantity representing the smoothness of the signal. It has already been realized by many that the problem of estimating the smoothness is not sensible. What can then be inferred about the smoothness? The paper attempts to answer this question. We consider implications of our results to hypothesis testing about the smoothness and smoothness classification problem. The test statistic is based on the empirical Bayes approach, i.e., it is the marginalized maximum likelihood estimator of the smoothness parameter for an appropriate prior distribution on the unknown signal. 相似文献
6.
V. G. Alekseev 《Mathematical Notes》1972,12(5):808-811
We investigate statistical estimates of a probability density distribution function and its derivatives. As the starting point of the investigation we take a priori assumptions about the degree of smoothness of the probability density to be estimated. By using these assumptions we can construct estimates of the probability density function itself and its derivatives which are distinguished by the high rate of decrease of the error in the estimate as the sample size increases.Translated from Matematicheskie Zametki, Vol. 12, No. 5, 621–626, November, 1972. 相似文献
7.
This survey paper studies the approximation of (polynomial) processes for which the operator norms do not form a bounded sequence. In view of familiar direct estimates and quantitative uniform boundedness principles, a unified approach is given to results concerning the equivalence of Dini-Lipschitz-type conditions with (strong) convergence on (smoothness) classes. Emphasis is laid upon the necessity of these conditions, essential ingredients of the proofs are suitable modifications of the familiar gliding hump method. Apart from the classical results concerned with Fourier partial sums, explicit applications are treated for (trigonometric as well as algebralc) Lagrange interpolation, interpolatory quadrature rules based upon Jacobl knots, multipliers or strong convergence, and for Bochner-Riesz means of multivariate Fourier series for parameter values below the critical index. 相似文献
8.
I. V. Smazhenko 《Ukrainian Mathematical Journal》2005,57(3):481-508
We consider a continuous function that changes its sign on an interval finitely many times and pose the problem of the approximation
of this function by a polynomial that inherits its sign. For this approximation, we obtain (in the case where this is possible)
Jackson-type estimates containing modified weighted moduli of smoothness of the Ditzian-Totik type. In some cases, constants
in these estimates depend substantially on the location of points where the function changes its sign. We give examples of
functions for which these constants are unimprovable. We also prove theorems that are analogous, in a certain sense, to inverse
theorems of approximation without restrictions.
__________
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 400–420, March, 2005. 相似文献
9.
The subject matter of this paper is an initial-value problem with an initial function for a linear differential difference equation of neutral type. The problem is to find an initial function such that the solution generated by this function has some given smoothness at the points multiple of the delay. The problem is solved using a method of polynomial quasisolutions, which is based on a representation of the unknown function in the form of a polynomial of some degree. Substituting this into the initial problem yields some incorrectness in the sense of degree of polynomials, which is compensated for by introducing some residual into the equation. For this residual, an exact analytical formula as a measure of disturbance of the initial-value problem is obtained. It is shown that if a polynomial quasisolution of degree N is chosen as an initial function for the initial-value problem in question, the solution generated will have smoothness not lower than N at the abutment points. 相似文献
10.
In this study, we consider the Bayesian estimation of unknown parameters and reliability function of the generalized exponential distribution based on progressive type-I interval censoring. The Bayesian estimates of parameters and reliability function cannot be obtained as explicit forms by applying squared error loss and Linex loss functions, respectively; thus, we present the Lindley’s approximation to discuss these estimations. Then, the Bayesian estimates are compared with the maximum likelihood estimates by using the Monte Carlo simulations. 相似文献
11.
Makio Ishiguro Yosiyuki Sakamoto 《Annals of the Institute of Statistical Mathematics》1984,36(1):523-538
Summary A Bayesian procedure for the probability density estimation is proposed. The procedure is based on the multinomial logit transformations
of the parameters of a finely segmented histogram model. The smoothness of the estimated density is guaranteed by the introduction
of a prior distribution of the parameters. The estimates of the parameters are defined as the mode of the posterior distribution.
The prior distribution has several adjustable parameters (hyper-parameters), whose values are chosen so that ABIC (Akaike's
Bayesian Information Criterion) is minimized.
The basic procedure is developed under the assumption that the density is defined on a bounded interval. The handling of the
general case where the support of the density function is not necessarily bounded is also discussed. The practical usefulness
of the procedure is demonstrated by numerical examples.
The Institute of Statistical Mathematics 相似文献
12.
本文对于单位球面上的经典连续模,给出了一个非常有用的广义Ul'yanov型不等式.该不等式在球面多项式逼近、球面嵌入理论以及球面上函数空间的插值理论等领域有着非常重要的应用.我们的证明基于球面调和多项式展开的新的估计,这些估计本身也具有独立的意义. 相似文献
13.
Yafang Gong 《分析论及其应用》2006,22(4):329-338
Based on Bernstein's Theorem, Kalandia's Lemma describes the error estimate and the smoothness of the remainder under the second part of Hoelder norm when a HSlder function is approximated by its best polynomial approximation. In this paper, Kalandia's Lemma is generalized to the cases that the best polynomial is replaced by one of its four kinds of Chebyshev polynomial expansions, the error estimates of the remainder are given out under Hoeder norm or the weighted HSlder norms. 相似文献
14.
In this note, we establish a new formulation of smoothness conditions for piecewise polynomial (: =pp) functions in terms
of the B-net representation in the general n-dimensional setting. It plays an important role for 2-dimensional setting in
the constructive proof of the fact that the spaces of polynomial splines with smoothness r and total degree k≥3r+2 over arbitrary
triangulations achieve the optimal approximation order with the approximation constant depending only on k and the smallest
angle of the partition in [5]. 相似文献
15.
随机加权法在密度估计中的应用 总被引:2,自引:0,他引:2
本文给出了概率密度函数的椭机加权估计,证明了承机加权分布与密度估计的标准化估计量的分布的逼近精度可达到o(1/√nh),并且构造了Efn(x)的置信区间,其中fn(x)为密度函数的核估计,h=hn炒估计的窗宽。 相似文献
16.
S. Le Borne 《PAMM》2003,2(1):21-24
Hierarchical matrices (ℋ︁‐matrices) provide a technique for the sparse approximation of large, fully populated matrices. This technique has been shown to be applicable to stiffness matrices arising in boundary element method applications where the kernel function displays certain smoothness properties. The error estimates for an approximation of the kernel function by a separable function can be carried over directly to error estimates for an approximation of the stiffness matrix by an ℋ︁‐matrix, using a certain standard partitioning and admissibility condition for matrix blocks. Similarly, ℋ︁‐matrix techniques can be applied in the finite element context where it is the inverse of the stiffness matrix that is fully populated. Here one needs a separable approximation of Green's function of the underlying boundary value problem in order to prove approximability by matrix blocks of low rank. Unfortunately, Green's function for the convection‐diffusion equation does not satisfy the required smoothness properties, hence prohibiting a straightforward generalization of the separable approximation through Taylor polynomials. We will use Green's function to motivate a modification in the (hierarchical) partitioning of the index set and as a consequence the resulting hierarchy of block partitionings as well as the admissibility condition. We will illustrate the effect of the proposed modifications by numerical results. 相似文献
17.
We consider 3-monotone approximation by piecewise polynomials with prescribed knots. A general theorem is proved, which reduces the problem of 3-monotone uniform approximation of a 3-monotone function, to convex local L1 approximation of the derivative of the function. As the corollary we obtain Jackson-type estimates on the degree of 3-monotone approximation by piecewise polynomials with prescribed knots. Such estimates are well known for monotone and convex approximation, and to the contrary, they in general are not valid for higher orders of monotonicity. Also we show that any convex piecewise polynomial can be modified to be, in addition, interpolatory, while still preserving the degree of the uniform approximation. Alternatively, we show that we may smooth the approximating piecewise polynomials to be twice continuously differentiable, while still being 3-monotone and still keeping the same degree of approximation. 相似文献
18.
The broad class of extended real-valued lower semicontinuous (lsc) functions on ? n captures nearly all functions of practical importance in equation solving, variational problems, fitting, and estimation. The paper develops piecewise polynomial functions, called epi-splines, that approximate any lsc function to an arbitrary level of accuracy. Epi-splines provide the foundation for the solution of a rich class of function identification problems that incorporate general constraints on the function to be identified including those derived from information about smoothness, shape, proximity to other functions, and so on. As such extrinsic information as well as observed function and subgradient values often evolve in applications, we establish conditions under which the computed epi-splines converge to the function we seek to identify. Numerical examples in response surface building and probability density estimation illustrate the framework. 相似文献
19.
Risk bounds for model selection via penalization 总被引:11,自引:0,他引:11
Andrew Barron Lucien Birgé Pascal Massart 《Probability Theory and Related Fields》1999,113(3):301-413
Performance bounds for criteria for model selection are developed using recent theory for sieves. The model selection criteria
are based on an empirical loss or contrast function with an added penalty term motivated by empirical process theory and roughly
proportional to the number of parameters needed to describe the model divided by the number of observations. Most of our examples
involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression
function. We show that the quadratic risk of the minimum penalized empirical contrast estimator is bounded by an index of the accuracy of the sieve. This accuracy index quantifies the trade-off among the candidate models
between the approximation error and parameter dimension relative to sample size.
If we choose a list of models which exhibit good approximation properties with respect to different classes of smoothness,
the estimator can be simultaneously minimax rate optimal in each of those classes. This is what is usually called adaptation. The type of classes of smoothness in which one gets adaptation depends heavily on the list of models. If too many models
are involved in order to get accurate approximation of many wide classes of functions simultaneously, it may happen that the
estimator is only approximately adaptive (typically up to a slowly varying function of the sample size).
We shall provide various illustrations of our method such as penalized maximum likelihood, projection or least squares estimation.
The models will involve commonly used finite dimensional expansions such as piecewise polynomials with fixed or variable knots,
trigonometric polynomials, wavelets, neural nets and related nonlinear expansions defined by superposition of ridge functions.
Received: 7 July 1995 / Revised version: 1 November 1997 相似文献
20.
In recent years there have been various attempts at the
representations of {\mbox multivariate} signals such as images, which
outperform wavelets. As is well known, wavelets are not optimal in
that they do not take full advantage of the geometrical
regularities and singularities of the images. Thus these
approaches have been based on tracing curves of singularities and
applying bandlets, curvelets, ridgelets, etc., or allocating some weights to curves of
singularities like the Mumford–Shah functional and its
modifications. In the latter approach a function is approximated
on subdomains where it is smoother but there is a penalty in the
form of the total length (or other measurement) of the
partitioning curves. We introduce a combined measure of smoothness
of the function in several dimensions by augmenting its smoothness
on subdomains by the smoothness of the partitioning curves.
Also, it is known that classical smoothness spaces fail to
characterize approximation spaces corresponding to multivariate
piecewise polynomial nonlinear approximation. We show how the
proposed notion of smoothness can almost characterize these
spaces. The question whether the characterization proposed in this
work can be further simplified remains open. 相似文献