关于变化再生核希尔伯特空间在线算法的一个注记

本文研究了由高斯核的方差在算法中引起的一些误差,利用再生核的一些特殊性质对这些误差进行分析.这些误差在分析算法的收敛速度时起到了重要的作用.     

QUANTUM GAUSSIAN PROCESSES

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算子方程基本问题解的再生核稀疏表示

在一个Hilbert空间中通过内积核定义的线性算子对应一个自然的再生核Hilbert空间结构.本文将称其为H-HK结构.这个结构本身内蕴一个基方法,可以解答线性算子的若干最基本的问题,包括确定或刻画其值域空间、解算子方程及解Moore-Penrose伪-(广义-)逆算子问题.在对已存在结果的简要综述之后,本文的目的是建立H-HK结构下的预正交自适应Fourier分解(pre-orthogonal adaptive Fourier decomposition,POAFD)算法.在这个方法之下导出上述3个问题的解的稀疏表示.在逐次跟踪匹配的优化方法论中POAFD的优选原理保证了它在理论上和实用上的最优性.它也具有算法上的可行性.所提供的方法可有效地应用于具体实际问题,包括信号与图像重构、常微分方程、偏微分方程和优化问题的数值解等.     

再生核希尔伯特空间连续线性泛函的范数及其应用

考虑了再生核希尔伯特空间连续线性泛函范数的表示,得到了用其范数平方等于该线性泛函连续两次作于再生核的简明表示.对于常见的Sobolev-Hibert空间而言,其再生核则可用截幂函数来表示,从而得到Sobolev-Hibert空间上连续线性泛函范数的简洁表示,以新视角解释和简化了文献中的现有结果.     

Convergence of probabilities for the second order monadic properties of a random mapping

We prove that the probability of each second order monadic property of a random mapping u_n converges as n→∞. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 277–295, 1997     

Robust designs for models with possible bias and correlated errors

This paper studies the model-robust design problem for general models with an unknown bias or contamination and the correlated errors. The true response function is assumed to be from a reproducing kernel Hilbert space and the errors are fitted by the qth order moving average process MA（q）, especially the MA（1） errors and the MA（2） errors. In both situations, design criteria are derived in terms of the average expected quadratic loss for the least squares estimation by using a minimax method. A case is studied and the orthogonality of the criteria is proved for this special response. The robustness of the design criteria is discussed through several numerical examples.     

Bergman空间上的复合算子的范数

本文研究了Bergman空间上的复合算子的范数与再生核的关系,证明了紧复合算子C的范数‖C‖=sup{‖C*kw‖:w∈D}的充要条件是(0)=0或是仿射映射,即(z)=sz+t,s,t是满足|s|+|t|<1的常数,其中kw为Bergman空间的规范再生核, C*是C的共轭算子.     

On Bohr's Inequality

Bohr's inequality says that if is a bounded analytic function on the closed unit disc, then for 0 leq r 1/3 and that1/3 is sharp. In this paper we give an operator-theoretic proofof Bohr's inequality that is based on von Neumann's inequality.Since our proof is operator-theoretic, our methods extend toseveral complex variables and to non-commutative situations. We obtain Bohr type inequalities for the algebras of boundedanalytic functions and the multiplier algebras of reproducingkernel Hilbert spaces on various higher-dimensional domains,for the non-commutative disc algebra A_n, and for the reduced(respectively full) group C*-algebra of the free group on ngenerators. We also include an application to Banach algebras. We provethat every Banach algebra has an equivalent norm in which itsatisfies a non-unital version of von Neumann's inequality. 2000 Mathematical Subject Classification: 47A20, 47A56.     

The gradient iteration for approximation in reproducing kernel Hilbert spaces

Consider the bounded linear operator, L: F Z, where Z R^N andF are Hilbert spaces defined on a common field X. L is madeup of a series of N bounded linear evaluation functionals, L_i:F R. By the Riesz representation theorem, there exist functionsk(x_i, ^·) F : L_if = f, k(x_i, ^·)_F. The functions,k(x_i, ^·), are known as reproducing kernels and F is areproducing kernel Hilbert space (RKHS). This is a natural frameworkfor approximating functions given a discrete set of observations.In this paper the computational aspects of characterizing suchapproximations are described and a gradient method presentedfor iterative solution. Such iterative solutions are desirablewhen N is large and the matrix computations involved in thebasic solution become infeasible. This is also exactly the casewhere the problem becomes ill-conditioned. An iterative approachto Tikhonov regularization is therefore also introduced. Unlikeiterative solutions for the more general Hilbert space setting,the proofs presented make use of the spectral representationof the kernel.     

Berezin Symbols and Schatten--von Neumann Classes

In terms of Berezin symbols, we give several criteria for operators to belong to the Schatten--von Neumann classes . In particular, for functions of model operators, we give a complete answer to a question posed by Nordgren and Rosenthal.     

度量空间中的软间隔分类

本文研究了度量空间中的软间隔分类问题,利用度量d的特性,得到一非线性映射,将度量空间等距嵌入Banach空间中,并构造了一种软间隔分类算法.     

Weighted L quantile distance estimators for randomly censored data

The asymptotic properties of a family of minimum quantile distance estimators for randomly censored data sets are considered. These procedures produce an estimator of the parameter vector that minimizes a weighted L² distance measure between the Kaplan-Meier quantile function and an assumed parametric family of quantile functions. Regularity conditions are provided which insure that these estimators are consistent and asymptotically normal. An optimal weight function is derived for single parameter families, which, for location/scale families, results in censored sample analogs of estimators such as those suggested by Parzen.