Conditional quantiles with varying Gaussians

In this paper we study conditional quantile regression by learning algorithms generated from Tikhonov regularization schemes associated with pinball loss and varying Gaussian kernels. Our main goal is to provide convergence rates for the algorithm and illustrate differences between the conditional quantile regression and the least square regression. Applying varying Gaussian kernels improves the approximation ability of the algorithm. Bounds for the sample error are achieved by using a projection operator, a variance-expectation bound derived from a condition on conditional distributions and a tight bound for the covering numbers involving the Gaussian kernels.     

Representing Superoscillations and Narrow Gaussians with Elementary Functions

A simple addition to the collection of superoscillatory functions is constructed, in the form of a square-integrable sinc function which is band-limited yet in some intervals oscillates faster than its highest Fourier component. Two parameters enable tuning of the local frequency of the superoscillations and the length of the interval over which they occur. Away from the superoscillatory intervals, the function rises to exponentially large values. An integral transform generates other band-limited functions with arbitrarily narrow peaks that are locally Gaussian. In the (delicate) limit of zero width, these would be Dirac delta-functions, which by superposition could enable construction of band-limited functions with arbitrarily fine structure.     

Exponential Estimates for an Empirical Bayes Risk in Classification of a Mixture with Varying Concentrations

For data obtained by sampling from a mixture of several components whose concentration varies in the the course of observations, we construct an empirical Bayes classifier and consider its asymptotic properties.     

On Euler products and multi-variate Gaussians

In this Note, we extend a recent result of A. Selberg concerning the asymptotic value distribution of Euler products to a multi-dimensional setting. Under certain conditions, an asymptotic development of Edgeworth type is found. To cite this article: D.A. Hejhal, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Learning and approximation by Gaussians on Riemannian manifolds

Learning function relations or understanding structures of data lying in manifolds embedded in huge dimensional Euclidean spaces is an important topic in learning theory. In this paper we study the approximation and learning by Gaussians of functions defined on a d-dimensional connected compact C ^∞ Riemannian submanifold of which is isometrically embedded. We show that the convolution with the Gaussian kernel with variance σ provides the uniform approximation order of O(σ ^s ) when the approximated function is Lipschitz s ∈(0, 1]. The uniform normal neighborhoods of a compact Riemannian manifold play a central role in deriving the approximation order. This approximation result is used to investigate the regression learning algorithm generated by the multi-kernel least square regularization scheme associated with Gaussian kernels with flexible variances. When the regression function is Lipschitz s, our learning rate is (log² m)/m) ^{s/(8 s + 4 d)} where m is the sample size. When the manifold dimension d is smaller than the dimension n of the underlying Euclidean space, this rate is much faster compared with those in the literature. By comparing approximation orders, we also show the essential difference between approximation schemes with flexible variances and those with a single variance. Supported partially by the Research Grants Council of Hong Kong [Project No. CityU 103405], City University of Hong Kong [Project No. 7001983], National Science Fund for Distinguished Young Scholars of China [Project No. 10529101], and National Basic Research Program of China [Project No. 973-2006CB303102].     

Bernstein-Type Inequalities for Linear Combinations of Shifted Gaussians

Let P_n be the collection of all polynomials of degree at mostn with real coefficients. A subtle Bernstein-type extremal problemis solved by establishing the inequality for all , and m= 1, 2, ..., where c is an absolute constant and Some related inequalities and direct and inversetheorems about the approximation by elements of in L_q (R) are also discussed. 2000 Mathematics SubjectClassification 41A17 (primary).     

Rational Approximation with Varying Weights III

Approximation by weighted rationals of the form wⁿr_n, where r_n=p_n/q_n, p_n and q_n are polynomials of degree at most [αn] and [βn], respectively, and w is an admissible weight, is investigated on compact subsets of the real line for a general class of weights and given α0, β0, with α+β>0. Conditions that characterize the largest sets on which such approximation is possible are given. We apply the general theorems to Laguerre and Freud weights.     

Location Among Regions with Varying Norms

This paper investigates a new variation in the continuous single facility location problem. Specifically, we address the problem of locating a new facility on a plane with different distance norms on different sides of a boundary line. Special cases and extensions of the problem, where there are more than two regions are also discussed. Finally, by investigating the properties of the models, efficient solution procedures are proposed.     

Asymptotics for Christoffel Functions with Varying Weights

We prove sharp asymptotics for Christoffel functions λ_n(w²ⁿ, · ) with respect to varying weights w²ⁿ. The result has an interpretation in the theory of statistical-mechanical models of quantum systems, and it is sufficiently strong to yield asymptotics for Christoffel functions for weights defined on several intervals.     

Scoring Unusual Words with Varying Mismatch Errors

Patterns consisting of strings with a bounded number of mismatches are central to coding theory and find multiple applications in text processing and computational biology. In this latter field, the presence of over-represented patterns of this kind has been linked, for instance, to modeling regulatory regions in biosequences. The study and computation of expected number of occurrences and related scores for these patterns is made difficult by the sheer explosion of the roster of candidates that need to be evaluated. In recent work, properties of pattern saturation and score monotonicity have proved capable to mitigate this problem. In such a context, expectation and score monotonicity has been established within the i.i.d. model for all cases of interest except that of a fixed word length with a varying number of mismatches. The present paper completes this investigation by showing that the expected number of occurrences in a textstring for such a word is bi-tonic, that is, behaves as a unimodal function of the number of errors. This extends to this case the time and space savings brought about by discovery algorithms based on pattern saturation. Work Supported in part by the Italian Ministry of University and Research under the Bi-National Project FIRB RBIN04BYZ7, and by the Research Program of Georgia Tech. An extended abstract related to this work was presented at the Dagstuhl Seminar Dagstuhl on “Combinatorial and Algorithmic Foundations of Pattern and Association Discovery”, May 14-19, 2006 [3].     

Multivariate Gaussians,semidefinite matrix completion,and convex algebraic geometry

We study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix. Maximum likelihood estimation for such models leads to the problem of maximizing the determinant function over a spectrahedron, and to the problem of characterizing the image of the positive definite cone under an arbitrary linear projection. These problems at the interface of statistics and optimization are here examined from the perspective of convex algebraic geometry.     

Static and Dynamic Layout Problems with Varying Areas

This paper describes a two-step algorithm for solving the layout problem while assuming the departments can have varying areas. The first step solves a quadratic assignment problem formulation of the problem using a heuristic cutting plane routine. The second step solves a mixed-integer linear programming prob- lem to find the desired block diagram layout. The algorithm incorporates two concepts to make the solu- tions more practical. First, rearrangement costs are simultaneously considered along with flow costs in solving a dynamic layout problem involving multiple time periods. It is the only algorithm to solve a general dynamic layout problem with varying department areas. Second, regular department shapes are maintained by requiring all departments to be rectangular. Its formulation for doing this is more efficient than previous algorithms.     

Online scheduling with reassignment

This paper studies online scheduling problems with reassignment on two identical machines. We can reassign some jobs under certain rules after all the jobs have been assigned. Three different versions are studied and optimal algorithms are proposed.     

缓变厚度中厚板的自由振动

本文将中厚板的厚度函数按一小参数展开,并采用奇异摄动方法,把原来变系数的微分方程组化成一系列常系数微分方程组求解.文中给出了任意变厚度中厚板的自振频率计算显式表达式,由此式,我们不仅可以方便地计算出各种变厚度的自振频率值,而且也可以根据频率的要求来优化板的厚度.文中的算例表明,本文的方法具有较好的精度、方法简便、有效等其他优点,可以考虑作为分析各种变厚度板壳的振动及稳定特征问题的有效方法之一.     

具变时滞细胞神经网络的周期解

利用拓扑度理论中的Mawhin延拓定理,得到了具变时滞细胞神经网络的周期解的充分条件,并改进推广了已有文献中的相应结论。     

Singularly Perturbed Nonlinear Integrodifferential Systems with Fast Varying Kernels

We consider nonlinear singularly perturbed integro-differential equations with fast varying kernels. It is assumed that the spectrum of the limiting operator lies in the closed left half-plane Re0. We derive an algorithm for obtaining regularized (in the sense of Lomov) asymptotic solutions in both the nonresonance and resonance cases. In deriving the algorithm, we essentially use the regularization apparatus for integral operators with fast varying kernels, developed earlier by the authors for linear integral and integro-differential systems. The algorithm is justified and the existence of a solution of the original nonlinear problem is proved by means of the Newton method for operator equations.     

适应性波束形成器的随机变界截尾算法

本文对适应性波束形成器给出一种递推随机变界截尾算法,并在较简单的条件下,证明了算法几乎处处具有大范围收敛性.     

Asymptotic Lipschitz Regularity for Tug-of-War Games with Varying Probabilities

We prove an asymptotic Lipschitz estimate for value functions of tug-of-war games with varying probabilities defined in Ω ? ?ⁿ. The method of the proof is based on a game-theoretic idea to estimate the value of a related game defined in Ω ×Ω via couplings.