Estimates of the norm of the Mercer kernel matrices with discrete orthogonal transforms

The paper is related to the lower and upper estimates of the norm for Mercer kernel matrices. We first give a presentation of the Lagrange interpolating operators from the view of reproducing kernel space. Then, we modify the Lagrange interpolating operators to make them bounded in the space of continuous function and be of the de la Vallée Poussin type. The order of approximation by the reproducing kernel spaces for the continuous functions is thus obtained, from which the lower and upper bounds of the Rayleigh entropy and the l ²-norm for some general Mercer kernel matrices are provided. As an example, we give the l ²-norm estimate for the Mercer kernel matrix presented by the Jacobi algebraic polynomials. The discussions indicate that the l ²-norm of the Mercer kernel matrices may be estimated with discrete orthogonal transforms. Supported by the national NSF (No: 10871226) of P.R. China.     

Derivative reproducing properties for kernel methods in learning theory

The regularity of functions from reproducing kernel Hilbert spaces (RKHSs) is studied in the setting of learning theory. We provide a reproducing property for partial derivatives up to order s when the Mercer kernel is C^2s. For such a kernel on a general domain we show that the RKHS can be embedded into the function space C^s. These observations yield a representer theorem for regularized learning algorithms involving data for function values and gradients. Examples of Hermite learning and semi-supervised learning penalized by gradients on data are considered.     

若干周期再生核空间的覆盖数

借助离散Fourier变换给出估计Mercer核矩阵逆矩阵范数上界的一种方法,由此给出了估计周期再生核Hilbert空间覆盖数的上、下界的一般方法.特别, 对两种特殊的周期再生核空间覆盖数的上、下界进行了比较.     

Reproducing Kernel Hilbert Spaces Associated with Analytic Translation-Invariant Mercer Kernels

In this article we study reproducing kernel Hilbert spaces (RKHS) associated with translation-invariant Mercer kernels. Applying a special derivative reproducing property, we show that when the kernel is real analytic, every function from the RKHS is real analytic. This is used to investigate subspaces of the RKHS generated by a set of fundamental functions. The analyticity of functions from the RKHS enables us to derive some estimates for the covering numbers which form an essential part for the analysis of some algorithms in learning theory. The work is supported by City University of Hong Kong (Project No. 7001816), and National Science Fund for Distinguished Young Scholars of China (Project No. 10529101).     

Geometry on Probability Spaces

Partial differential equations and the Laplacian operator on domains in Euclidean spaces have played a central role in understanding natural phenomena. However, this avenue has been limited in many areas where calculus is obstructed, as in singular spaces, and in function spaces of functions on a space X where X itself is a function space. Examples of the latter occur in vision and quantum field theory. In vision it would be useful to do analysis on the space of images and an image is a function on a patch. Moreover, in analysis and geometry, the Lebesgue measure and its counterpart on manifolds are central. These measures are unavailable in the vision example and even in learning theory in general. There is one situation where, in the last several decades, the problem has been studied with some success. That is when the underlying space is finite (or even discrete). The introduction of the graph Laplacian has been a major development in algorithm research and is certainly useful for unsupervised learning theory. The approach taken here is to take advantage of both the classical research and the newer graph theoretic ideas to develop geometry on probability spaces. This starts with a space X equipped with a kernel (like a Mercer kernel) which gives a topology and geometry; X is to be equipped as well with a probability measure. The main focus is on a construction of a (normalized) Laplacian, an associated heat equation, diffusion distance, etc. In this setting, the point estimates of calculus are replaced by integral quantities. One thinks of secants rather than tangents. Our main result bounds the error of an empirical approximation to this Laplacian on X.     

Learning gradients by a gradient descent algorithm

We propose a stochastic gradient descent algorithm for learning the gradient of a regression function from random samples of function values. This is a learning algorithm involving Mercer kernels. By a detailed analysis in reproducing kernel Hilbert spaces, we provide some error bounds to show that the gradient estimated by the algorithm converges to the true gradient, under some natural conditions on the regression function and suitable choices of the step size and regularization parameters.     

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.     

非齐型空间上Triebel-Lizorkin空间的T1定理

本文在非齐型空间上证明具有Dini核条件的T1定理,获得了加权Fefferman- Stein向量值极大不等式．进一步地,在非齐型空间上得到了加权Tiebel-Lizorkin空间的T1定理．     

Frames of Poisson wavelets on the sphere

In this paper we show that for sufficiently dense grids Poisson wavelets on the sphere constitute a weighted frame. In the proof we will only use the localization properties of the reproducing kernel and its gradient. This indicates how this kind of theorem can be generalized to more general reproducing kernel Hilbert spaces. With the developed technique we prove a sampling theorem for weighted Bergman spaces.     

The covering number in learning theory

The covering number of a ball of a reproducing kernel Hilbert space as a subset of the continuous function space plays an important role in Learning Theory. We give estimates for this covering number by means of the regularity of the Mercer kernel K. For convolution type kernels K(x,t)=k(x−t) on [0,1]ⁿ, we provide estimates depending on the decay of k̂, the Fourier transform of k. In particular, when k̂ decays exponentially, our estimate for this covering number is better than all the previous results and covers many important Mercer kernels. A counter example is presented to show that the eigenfunctions of the Hilbert–Schmidt operator Lm_K associated with a Mercer kernel K may not be uniformly bounded. Hence some previous methods used for estimating the covering number in Learning Theory are not valid. We also provide an example of a Mercer kernel to show that L_K^1/2 may not be generated by a Mercer kernel.     

The Liouville’s theorem of harmonic functions on Alexandrov spaces with nonnegative Ricci curvature

In this paper, the authors prove the Liouville’s theorem for harmonic function on Alexandrov spaces by heat kernel approach, which extends the Liouville’s theorem of harmonic function from Riemannian manifolds to Alexandrov spaces.     

Mercer’s Theorem on General Domains: On the Interaction between Measures, Kernels, and RKHSs

Given a compact metric space X and a strictly positive Borel measure ν on X, Mercer’s classical theorem states that the spectral decomposition of a positive self-adjoint integral operator T _k :L ₂(ν)→L ₂(ν) of a continuous k yields a series representation of k in terms of the eigenvalues and -functions of T _k . An immediate consequence of this representation is that k is a (reproducing) kernel and that its reproducing kernel Hilbert space can also be described by these eigenvalues and -functions. It is well known that Mercer’s theorem has found important applications in various branches of mathematics, including probability theory and statistics. In particular, for some applications in the latter areas, however, it would be highly convenient to have a form of Mercer’s theorem for more general spaces X and kernels k. Unfortunately, all extensions of Mercer’s theorem in this direction either stick too closely to the original topological structure of X and k, or replace the absolute and uniform convergence by weaker notions of convergence that are not strong enough for many statistical applications. In this work, we fill this gap by establishing several Mercer type series representations for k that, on the one hand, make only very mild assumptions on X and k, and, on the other hand, provide convergence results that are strong enough for interesting applications in, e.g., statistical learning theory. To illustrate the latter, we first use these series representations to describe ranges of fractional powers of T _k in terms of interpolation spaces and investigate under which conditions these interpolation spaces are contained in L _∞(ν). For these two results, we then discuss applications related to the analysis of so-called least squares support vector machines, which are a state-of-the-art learning algorithm. Besides these results, we further use the obtained Mercer representations to show that every self-adjoint nuclear operator L ₂(ν)→L ₂(ν) is an integral operator whose representing function k is the difference of two (reproducing) kernels.     

On an intertwining lifting theorem for certain reproducing kernel Hilbert spaces

We provide an alternate approach to an intertwining lifting theorem obtained by Ball, Trent and Vinnikov. The results are an exact analogue of the classical Sz-Nagy-Foias theorem in the case of multipliers on a class of reproducing kernel spaces, which satisfy the Nevanlinna-Pick property.     

Weighted Fourier–Laplace transforms in reproducing kernel Hilbert spaces on the sphere

We study the action of a weighted Fourier–Laplace transform on the functions in the reproducing kernel Hilbert space (RKHS) associated with a positive definite kernel on the sphere. After defining a notion of smoothness implied by the transform, we show that smoothness of the kernel implies the same smoothness for the generating elements (spherical harmonics) in the Mercer expansion of the kernel. We prove a reproducing property for the weighted Fourier–Laplace transform of the functions in the RKHS and embed the RKHS into spaces of smooth functions. Some relevant properties of the embedding are considered, including compactness and boundedness. The approach taken in the paper includes two important notions of differentiability characterized by weighted Fourier–Laplace transforms: fractional derivatives and Laplace–Beltrami derivatives.     

The convergence rates of Shannon sampling learning algorithms

In the present paper,we provide an error bound for the learning rates of the regularized Shannon sampling learning scheme when the hypothesis space is a reproducing kernel Hilbert space(RKHS) derived by a Mercer kernel and a determined net.We show that if the sample is taken according to the determined set,then,the sample error can be bounded by the Mercer matrix with respect to the samples and the determined net.The regularization error may be bounded by the approximation order of the reproducing kernel Hilbert space interpolation operator.The paper is an investigation on a remark provided by Smale and Zhou.     

Continuity and Schatten–von Neumann Properties for Localization Operators on Modulation Spaces

We use sharp convolution estimates for weighted Lebesgue and modulation spaces to obtain an extension of the celebrated Cordero-Gröchenig theorems on boundedness and Schatten–von Neumann properties of localization operators on modulation spaces. We also give a new proof of the Weyl connection based on the kernel theorem for Gelfand–Shilov spaces.     

非紧致一秩Riemann对称空间上的中心极限定理

我们在本文中研究非紧致一秩Ｒｉｅｍａｎｎ对称空间上初等球函数的渐近表示,并利用Ｌｏｈｏｕｅ Ｎ．和Ｒｙｃｈｎｅｒ Ｔｈ．得到的热核表达式,建立起这类空间上的非欧中心极限定理,所得结果包含了Ｔｅｒｒａｓ的定理作为其特例．     

Reproducing kernel method to solve parabolic partial differential equations with nonlocal conditions

In this study, the parabolic partial differential equations with nonlocal conditions are solved. To this end, we use the reproducing kernel method (RKM) that is obtained from the combining fundamental concepts of the Galerkin method, and the complete system of reproducing kernel Hilbert space that was first introduced by Wang et al. who implemented RKM without Gram–Schmidt orthogonalization process. In this method, first the reproducing kernel spaces and their kernels such that satisfy the nonlocal conditions are constructed, and then the RKM without Gram–Schmidt orthogonalization process on the considered problem is implemented. Moreover, convergence theorem, error analysis theorems, and stability theorem are provided in detail. To show the high accuracy of the present method several numerical examples are solved.     

Correction to ``The ghost of an index theorem'

The ``ghost of an index theorem" is an isomorphism between products of the kernel spaces and the cokernel spaces of a pair of bounded operators and their product, valid when each operator and also the product is assumed to have a generalized inverse. In this note we correct an error in the original proof, and extend the result to operators with closed range.