装备费用预测中的混合核函数支持向量机

在支持向量机预测建模中,核函数用来将低维特征空间中的非线性问题映射为高维特征空间中的线性问题.核函数的特征对于支持向量机的学习和预测都有很重要的影响.考虑到两种典型核函数—全局核(多项式核函数)和局部核(RBF核函数)在拟合与泛化方面的特性,采用了一种基于混合核函数的支持向量机方法用于预测建模.为了评价不同核函数的建模效果、得到更好的预测性能,采用遗传算法自适应进化支持向量机模型的各项参数,并将其应用于装备费用预测的实际问题中.实际计算表明采用混合核函数的支持向量机较单一核函数时有更好的预测性能,可以作为一种有效的预测建模方法在装备管理中推广应用.     

再生核与小波变换

在再生核基本理论的基础上,介绍了再生核在小波变换中的作用,并且根据连续小波变换像空间是再生核Hilbert空间这一基本事实,借助再生核理论的特殊技巧,建立了Littlewood-Paley和Haar小波变换像空间的再生核函数与已知再生核空间的再生核的关系,为小波变换像空间的进一步研究提供理论基础.     

Analyse sur les groupes de Lie à croissance polynômiale

From Gaussian upper bounds on the heat kernel we deduce similar upper bounds on the first space derivatives of the heat kernel. Gaussian lower bounds on the heat kernel are deduced and some applications are given.     

基于高斯核支持向量机的商品期货市场套利研究

基于高斯RBF核支持向量机预测棉花商品期货主力和次主力合约协整关系的价差序列,确定最优SVM参数,并选择合适的开平仓阈值,进行同品种跨期套利.再与多项式核支持向量机套利结果对比,得到在所有开平仓阈值上,基于高斯RBF核支持向量机套利的收益率都明显高于多项式核支持向量机套利的收益率.     

Solving integro-differential equations with Cauchy kernel

A simple method for solving the Fredholm singular integro-differential equations with Cauchy kernel is proposed based on a new reproducing kernel space. Using a transformation and modifying the traditional reproducing kernel method, the singular term is removed and the analytical representation of the exact solution is obtained in the form of series in the new reproducing kernel space. The advantage of the approach lies in the fact that, on the one hand, by improving the definition of traditional inner product, the representation of new reproducing kernel function becomes simple and requirement for image space of operator is weakened comparing with traditional reproducing kernel method; on the other hand, the approximate solution and its derivatives converge uniformly to the exact solution and its derivatives. Some examples are displayed to demonstrate the validity and applicability of the proposed method.     

Applications of the Bernstein-Durrmeyer operators in estimating the norm of Mercer kernel matrices

The paper is related to the norm estimate of Mercer kernel matrices.The lower and upper bound estimates of Rayleigh entropy numbers for some Mercer kernel matrices on[0,1]×[0,1]based on the Bernstein-Durrmeyer operator kernel ale obtained,with which and the approximation property of the Bernstein-Durrmeyer operator the lower and upper bounds of the Rayleigh entropy number and the l2-norm for general Mercer kernel matrices on[0,1]×[0,1]are provided.     

About the non-convex optimization problem induced by non-positive semidefinite kernel learning

During the last years, kernel based methods proved to be very successful for many real-world learning problems. One of the main reasons for this success is the efficiency on large data sets which is a result of the fact that kernel methods like support vector machines (SVM) are based on a convex optimization problem. Solving a new learning problem can now often be reduced to the choice of an appropriate kernel function and kernel parameters. However, it can be shown that even the most powerful kernel methods can still fail on quite simple data sets in cases where the inherent feature space induced by the used kernel function is not sufficient. In these cases, an explicit feature space transformation or detection of latent variables proved to be more successful. Since such an explicit feature construction is often not feasible for large data sets, the ultimate goal for efficient kernel learning would be the adaptive creation of new and appropriate kernel functions. It can, however, not be guaranteed that such a kernel function still leads to a convex optimization problem for Support Vector Machines. Therefore, we have to enhance the optimization core of the learning method itself before we can use it with arbitrary, i.e., non-positive semidefinite, kernel functions. This article motivates the usage of appropriate feature spaces and discusses the possible consequences leading to non-convex optimization problems. We will show that these new non-convex optimization SVM are at least as accurate as their quadratic programming counterparts on eight real-world benchmark data sets in terms of the generalization performance. They always outperform traditional approaches in terms of the original optimization problem. Additionally, the proposed algorithm is more generic than existing traditional solutions since it will also work for non-positive semidefinite or indefinite kernel functions.     

Covering numbers of isotropic reproducing kernels on compact two‐point homogeneous spaces

In this paper we present upper and lower estimates for the covering numbers of the unit ball of a reproducing kernel Hilbert space associated to a continuous isotropic kernel on a compact two‐point homogeneous space (CTPHS). These estimates are obtained from estimates on the decay of the Fourier–Jacobi coefficients of the kernel via applications of the Funk–Hecke formula and the Schoenberg series representation of an isotropic kernel on CTPHS and also by the use of cubature formulas on these spaces.     

Lévy flights over quantum paths

An impact of integration over the paths of the Lévy flights on the quantum mechanical kernel has been studied. Analytical expression for a free particle kernel has been obtained in terms of the Fox H-function. A new equation for the kernel of a particle in the box has been found. New general results include the well known quantum formulae for a free particle kernel and particle in box kernel.     

Sampling in reproducing kernel Banach spaces on Lie groups

We present sampling theorems for reproducing kernel Banach spaces on Lie groups. Recent approaches to this problem rely on integrability of the kernel and its local oscillations. In this paper, we replace these integrability conditions by requirements on the derivatives of the reproducing kernel and, in particular, oscillation estimates are found using derivatives of the reproducing kernel. This provides a convenient path to sampling results on reproducing kernel Banach spaces. Finally, these results are used to obtain frames and atomic decompositions for Banach spaces of distributions stemming from a cyclic representation. It is shown that this process is particularly easy, when the cyclic vector is a Gårding vector for a square integrable representation.     

Journe小波变换像空间描述

1引言1950年N.Aronszjan发表的一篇综述性文章《Theory of reproducing kernels》标志再生核理论的初步形成.由于再生核有许多良好的计算性质,S.Saitoh总结并深入研究再生核基本理论,进一步拓展了再生核的应用领域;徐利治把再生核应用于L~2(B)(B是复平面上的一个区域)中解析函数重积分降维问题,并提出了一个能对一些预先给出的     

计算再生核的Green函数方法

本文讨论了利用Green函数计算再生核的方法,在Wm2空间中利用再生核的和性质以及Green函数理论给出再生核构造的一般方法,并利用此方法计算出W32空间的再生核.     

Radial kernels and their reproducing kernel Hilbert spaces

We describe how to use Schoenberg’s theorem for a radial kernel combined with existing bounds on the approximation error functions for Gaussian kernels to obtain a bound on the approximation error function for the radial kernel. The result is applied to the exponential kernel and Student’s kernel. To establish these results we develop a general theory regarding mixtures of kernels. We analyze the reproducing kernel Hilbert space (RKHS) of the mixture in terms of the RKHS’s of the mixture components and prove a type of Jensen inequality between the approximation error function for the mixture and the approximation error functions of the mixture components.     

第二类端点奇异Fredholm积分方程的分数阶退化核方法

本文针对第二类端点奇异Fredholm积分方程构造基于分数阶Taylor展开的退化核方法，设计了两种计算格式，一是在全区间上使用分数阶Taylor展开式近似核函数，二是在包含奇点的小区间上采用分数阶插值，在剩余区间上采用分段二次多项式插值逼近核函数.讨论了两种退化核方法收敛的条件，并给出了混合插值法的收敛阶估计.数值算例表明对于非光滑核函数分数阶退化核方法有着良好的计算效果，且混合二次插值法比全区间上的分数阶退化核方法有着更广泛的适用范围.     

再生核移位勒让德基函数法求解分数阶微分方程

该文以再生核理论为基础,用移位Legendre多项式作为基函数构造了一个新的再生核空间,并给出了该空间下的再生核函数.与经典的再生核函数有所不同的是该空间下的再生核函数不再是分段函数,因此可以减小分数阶算子作用在核函数上时的计算量,使近似解更为精确.数值算例表明该方法的有效性.     

Reproducing Properties of Differentiable Mercer-Like Kernels on the Sphere

We study differentiability of functions in the reproducing kernel Hilbert space (RKHS) associated with a smooth Mercer-like kernel on the sphere. We show that differentiability up to a certain order of the kernel yields both, differentiability up to the same order of the elements in the series representation of the kernel and a series representation for the corresponding derivatives of the kernel. These facts are used to embed the RKHS into spaces of differentiable functions and to deduce reproducing properties for the derivatives of functions in the RKHS. We discuss compactness and boundedness of the embedding and some applications to Gaussian-like kernels.     

Equivariant heat invariants of the laplacian and nonmininmal operators on differential forms

In this paper,we compute the first two equivariant heat kernel coeffcients of the Bochner Laplacian on differential forms.The first two equivariant heat kernel coeffcients of the Bochner Laplacian with torsion are also given.We also study the equivariant heat kernel coeffcients of nonminimal operators on differential forms and get the equivariant Gilkey-Branson-Fulling formula.     

On the Kernel of the Equivariant Dirac Operator

We prove a vanishing theorem for certain isotypical components of the kernel of the S¹-equivariant Dirac operator with coefficients in an admissible Clifford module. The method is based on changing the metric by a conformal (generally unbounded) factor and studying the effect of this change on the Dirac operator and its kernel. In the cases relevant to S¹-actions we find that the kernel of the new operator is naturally isomorphic to the kernel of the original operator.     

Learning the optimal kernel for Fisher discriminant analysis via second order cone programming

Kernel Fisher discriminant analysis (KFDA) is a popular classification technique which requires the user to predefine an appropriate kernel. Since the performance of KFDA depends on the choice of the kernel, the problem of kernel selection becomes very important. In this paper we treat the kernel selection problem as an optimization problem over the convex set of finitely many basic kernels, and formulate it as a second order cone programming (SOCP) problem. This formulation seems to be promising because the resulting SOCP can be efficiently solved by employing interior point methods. The efficacy of the optimal kernel, selected from a given convex set of basic kernels, is demonstrated on UCI machine learning benchmark datasets.     

A Comparison Between the Bergman and Szegö Kernels of the Non-smooth Worm Domain $$D'_{\upbeta }$$

In this work we provide an asymptotic expansion for the Szegö kernel associated to a suitably defined Hardy space on the non-smooth worm domain \(D'_{\upbeta }\). After describing the singularities of the kernel, we compare it with an asymptotic expansion of the Bergman kernel. In particular, we show that the Bergman kernel has the same singularities of the first derivative of the Szegö kernel with respect to any of the variables. On the side, we prove the boundedness of the Bergman projection operator on Sobolev spaces of integer order.