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1.
We consider the problem of separating two sets of points in an n-dimensional real space with a (hyper)plane that minimizes the sum of L p -norm distances to the plane of points lying on the wrong side of it. Despite recent progress, practical techniques for the exact solution of cases other than the L 1 and L -norm were unavailable. We propose and implement a new approach, based on non-convex quadratic programming, for the exact solution of the L 2-norm case. We solve in reasonable computing times artificial problems of up to 20000 points (in 6 dimensions) and 13 dimensions (with 2000 points). We also observe that, for difficult real-life instances from the UCI Repository, computation times are substantially reduced by incorporating heuristic results in the exact solution process. Finally, we compare the classification performance of the planes obtained for the L 1, L 2 and L formulations. It appears that, despite the fact that L 2 formulation is computationally more expensive, it does not give significantly better results than the L 1 and L formulations.  相似文献   

2.
We determine the L p discrepancy of the two-dimensional Hammersley point set in base b. These formulas show that the L p discrepancy of the Hammersley point set is not of best possible order with respect to the general (best possible) lower bound on L p discrepancies due to Roth and Schmidt. To overcome this disadvantage we introduce permutations in the construction of the Hammersley point set and show that there always exist permutations such that the L p discrepancy of the generalized Hammersley point set is of best possible order. For the L 2 discrepancy such permutations are given explicitly. F.P. is supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.  相似文献   

3.
With the objective of generating “shape-preserving” smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based -smooth univariate cubic L 1 splines. An L 1 spline minimizes the L 1 norm of the difference between the first-order derivative of the spline and the local divided difference of the data. Calculating the coefficients of an L 1 spline is a nonsmooth non-linear convex program. Via Fenchel’s conjugate transformation, the geometric dual program is a smooth convex program with a linear objective function and convex cubic constraints. The dual-to-primal transformation is accomplished by solving a linear program.  相似文献   

4.
This work is concerned with the proof of Lp -Lq decay estimates for solutions of the Cauchy problem for utt -λ2(t)b2(t)/Δu =0. The coefficient consists of an increasing smooth function λ and an oscillating smooth and bounded function b which are uniformly separated from zero. The authors‘ main interest is devoted to the critical case where one has an interesting interplay between the growing and the oscillating part.  相似文献   

5.
The classes of the Lp,∞- and Lp-metrics play an important role to develop a probability theory in fuzzy sample spaces. All of these metrics are known to be separable, but not complete. The classes are closely related as for each Lp,∞-metric there exists some Lp-metric which induces the same topology. This paper deals with the completion of the Lp,∞- and Lp-metrics. We can also show that the relationship between the classes of Lp,∞- and Lp-metrics still holds for the obtained respective classes of their completions.  相似文献   

6.
We investigate Besov spaces and their connection with trigonometric polynomial approximation inL p[−π,π], algebraic polynomial approximation inL p[−1,1], algebraic polynomial approximation inL p(S), and entire function of exponential type approximation inL p(R), and characterizeK-functionals for certain pairs of function spaces including (L p[−π,π],B s a(L p[−π,π])), (L p(R),s a(Lp(R))), , and , where 0<s≤∞, 0<p<1,S is a simple polytope and 0<α<r. This project is supported by the National Science Foundation of China.  相似文献   

7.
We study the worst case setting for approximation of d variate functions from a general reproducing kernel Hilbert space with the error measured in the L norm. We mainly consider algorithms that use n arbitrary continuous linear functionals. We look for algorithms with the minimal worst case errors and for their rates of convergence as n goes to infinity. Algorithms using n function values will be analyzed in a forthcoming paper.We show that the L approximation problem in the worst case setting is related to the weighted L2 approximation problem in the average case setting with respect to a zero-mean Gaussian stochastic process whose covariance function is the same as the reproducing kernel of the Hilbert space. This relation enables us to find optimal algorithms and their rates of convergence for the weighted Korobov space with an arbitrary smoothness parameter α>1, and for the weighted Sobolev space whose reproducing kernel corresponds to the Wiener sheet measure. The optimal convergence rates are n-(α-1)/2 and n-1/2, respectively.We also study tractability of L approximation for the absolute and normalized error criteria, i.e., how the minimal worst case errors depend on the number of variables, d, especially when d is arbitrarily large. We provide necessary and sufficient conditions on tractability of L approximation in terms of tractability conditions of the weighted L2 approximation in the average case setting. In particular, tractability holds in weighted Korobov and Sobolev spaces only for weights tending sufficiently fast to zero and does not hold for the classical unweighted spaces.  相似文献   

8.
We derive a lower bound of L p norms, 1 ⩽ p ⩽ ∞, in the central limit theorem for strongly mixing random variables X 1,..., X n with under the boundedness condition ℙ{|X i | ⩽ M} = 1 with a nonrandom constantM > 0 and condition ∑ r⩾1 r 2α(r) < ∞, where α(r) are the Rosenblatt strong mixing coefficients. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 587–602, October–December, 2005.  相似文献   

9.
Let π and π′ be automorphic irreducible cuspidal representations of GLm(QA) and GLm(QA), respectively. Assume that π and π′ are unitary and at least one of them is self-contragredient. In this article we will give an unconditional proof of an orthogonality for π and π′, weighted by the von Mangoldt function Λ(n) and 1−n/x. We then remove the weighting factor 1−n/x and prove the Selberg orthogonality conjecture for automorphic L-functions L(s,π) and L(s,π′), unconditionally for m≤4 and m′≤4, and under the Hypothesis H of Rudnick and Sarnak [20] in other cases. This proof of Selberg's orthogonality removes such an assumption in the computation of superposition distribution of normalized nontrivial zeros of distinct automorphic L-functions by Liu and Ye [12].  相似文献   

10.
Lα (0 α 1) is a class of infinitely divisible distributions defined by restricting the measure in the Levy-Khinchin formula to a special form. When α = 1, Lα is just the classical class L. Several properties for Lα classes, which are similar to the most important properties for the class L, are established. Also, a conjecture of Wolfe about unimodality of some Lα distributions is disproved by giving a counterexample.  相似文献   

11.
In this paper the author first introduce a new concept of L p -dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of L p intersection body to L p -mixed intersection body. Inequalities for L p -dual mixed volumes of L p -mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities. This work was supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. Y605065) and the Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)  相似文献   

12.
In this paper, we present a correct proof of an L p -inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund’s inequality to the polar derivative of a polynomial.  相似文献   

13.
A robustified residual autocorrelation is defined based onL 1-regression. Under very general conditions, the asymptotic distribution of the robust residual autocorrelation is obtained. A robustified portmanteau statistic is then constructed which can be used in checking the goodness-of-fit of AR(p) models when usingL 1-norm fitting. Empirical results show thatL 1-norm estimators and the proposed portmanteau statistic are robust against outliers, error distributions, and accuracy for a given finite sample. Project supported by the Foundation of State Educational Commission and a research grant from the Doctoral Program Foundation of China (#97000139).  相似文献   

14.
We prove that the univariate interpolating cubic L 1 spline to the Heaviside function at three sites to the left of the jump and three sites to the right of the jump entirely agrees with the Heaviside function except in the middle interval where it is the interpolating cubic with zero slopes at the end point. This shows that there is no oscillation near the discontinuous point i.e. no Gibbs’ phenomenon.  相似文献   

15.
We prove local weighted integral inequalities for differential forms. Then byusing the local results, we prove global weighted integral inequalities for differential forms in L s (μ)-averaging domains and in John domains, respectively, which can be considered as generalizations of the classical Poincaré-type inequality.  相似文献   

16.
Associated with the L p -curvature image defined by Lutwak, some inequalities for extended mixed p-affine surface areas of convex bodies and the support functions of L p -projection bodies are established. As a natural extension of a result due to Lutwak, an L p -type affine isoperimetric inequality, whose special cases are L p -Busemann-Petty centroid inequality and L p -affine projection inequality, respectively, is established. Some L p -mixed volume inequalities involving L p -projection bodies are also established.  相似文献   

17.
In this paper we study the L p -discrepancy of digitally shifted Hammersley point sets. While it is known that the (unshifted) Hammersley point set (which is also known as Roth net) with N points has L p -discrepancy (p an integer) of order (log N)/N, we show that there always exists a shift such that the digitally shifted Hammersley point set has L p -discrepancy (p an even integer) of order which is best possible by a result of W. Schmidt. Further we concentrate on the case p = 2. We give very tight lower and upper bounds for the L 2-discrepancy of digitally shifted Hammersley point sets which show that the value of the L 2-discrepancy of such a point set mostly depends on the number of zero coordinates of the shift and not so much on the position of these. This work is supported by the Austrian Research Fund (FWF), Project P17022-N12 and Project S8305.  相似文献   

18.
The bivariate location problem is considered. The sup, L 1 and L 2 norms are used to construct bivariate sign tests from the univariate sign statistics computed on the projected observations on all lines passing through the origin. The tests so obtained are affine-invariant and distribution-free under the null hypothesis. The sup-norm gives rise to Hodges' test. A class of tests derived from the L 2-norm, with Blumen's test as a member, is seen to be related to a class proposed by Oja and Nyblom (1989, J. Amer. Statist. Assoc., 84, 249-259). The L 1-norm gives rise to a new test. Its asymptotic null distribution is seen to be the same as that of the L 1-norm of a certain normal process related to the standard Wiener process. An explicit expression of its cumulative distribution function is given. A simulation study will examine the merits of the three approaches.  相似文献   

19.
Weak versions of amenability components of BanachL 1(G)-modules are considered. Using these versions, a mean ergodic theorem for locally compact groups is formulated. The possibility of using weak amenability components of operatorL 1(G)-modules to characterize ameable groups is studied. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 879–886, December, 1999.  相似文献   

20.
郑绿洲  魏正理 《数学杂志》2014,34(4):617-626
本文研究了L_p球的相关问题.利用对偶混合体积、球面Radon变换和Fourier变换的方法,获得了关于L_p球的几个新不等式和性质,其中一个不等式与著名的最大切片猜想有关.  相似文献   

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