On spline approximation with a reproducing kernel method

Spline approximation with a reproducing kernel of a semi-Hilbert space is studied. Conditions are formulated that uniquely identify the natural Hilbert space by a reproducing kernel, a trend of the spline, and the approximation domain. The construction of a spline with external drift is proposed. It allows one to approximate functions having areas of large gradients or first-kind discontinuities. The conditional positive definiteness of some known radial basis functions is proved.     

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.     

Generalized Sobolev's phi function for the resolvent of a Milne-type integral equation with a degenerate kernel

It is well known in the field of radiative transfer that Sobolev was the first to introduce the resolvent into Milne's integral equation with a displacement kernel. Thereafter it was shown that the resolvent plays an important role in the theory of formation of spectral lines. In the theory of line-transfer problems, the kernel representation in Milne's integral equation has been used to provide an approximate solution in a manner similar to that given by the discrete ordinales method.In this paper, by means of invariant imbedding we show how to determine an exact solution of a Milne-type integral equation with a degenerate kernel, whose form is more general than the Pincherle-Gourast kernel. A Cauchy system for the resolvent is expressed in terms of generalized Sobolev's Φ- and Ψ-functions, which are computed by solving a system of differential equations for auxiliary functions. Furthermore, these functions are expressed in terms of components of the kernel representation.     

Approximation in Sobolev Spaces by Kernel Expansions

For interpolation of smooth functions by smooth kernels having an expansion into eigenfunctions (e.g., on the circle, the sphere, and the torus), good results including error bounds are known, provided that the smoothness of the function is closely related to that of the kernel. The latter fact is usually quantified by the requirement that the function should lie in the “native” Hilbert space of the kernel, but this assumption rules out the treatment of less smooth functions by smooth kernels. For the approximation of functions from “large” Sobolev spaces W by functions generated by smooth kernels, this paper shows that one gets at least the known order for interpolation with a less smooth kernel that has W as its native space.     

Solving the Dym initial value problem in reproducing kernel space

We consider two numerical solution approaches for the Dym initial value problem using the reproducing kernel Hilbert space method. For each solution approach, the solution is represented in the form of a series contained in the reproducing kernel space, and a truncated approximate solution is obtained. This approximation converges to the exact solution of the Dym problem when a sufficient number of terms are included. In the first approach, we avoid to perform the Gram-Schmidt orthogonalization process on the basis functions, and this will decrease the computational time. Meanwhile, in the second approach, working with orthonormal basis elements gives some numerical advantages, despite the increased computational time. The latter approach also permits a more straightforward convergence analysis. Therefore, there are benefits to both approaches. After developing the reproducing kernel Hilbert space method for the numerical solution of the Dym equation, we present several numerical experiments in order to show that the method is efficient and can provide accurate approximations to the Dym initial value problem for sufficiently regular initial data after relatively few iterations. We present the absolute error of the results when exact solutions are known and residual errors for other cases. The results suggest that numerically solving the Dym initial value problem in reproducing kernel space is a useful approach for obtaining accurate solutions in an efficient manner.     

Quasisymmetric Schur functions

We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to standard bases, also known as Demazure atoms. Our new basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions for quasisymmetric Schur functions in terms of monomial and fundamental quasisymmetric functions, which give rise to quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux. From here we derive a Pieri rule for quasisymmetric Schur functions that naturally refines the Pieri rule for Schur functions. After surveying combinatorial formulas for Macdonald polynomials, including an expansion of Macdonald polynomials into fundamental quasisymmetric functions, we show how some of our results can be extended to include the t parameter from Hall-Littlewood theory.     

Approximation on the sphere using radial basis functions plus polynomials

In this paper we analyse a hybrid approximation of functions on the sphere by radial basis functions combined with polynomials, with the radial basis functions assumed to be generated by a (strictly) positive definite kernel. The approximation is determined by interpolation at scattered data points, supplemented by side conditions on the coefficients to ensure a square linear system. The analysis is first carried out in the native space associated with the kernel (with no explicit polynomial component, and no side conditions). A more refined error estimate is obtained for functions in a still smaller space. Numerical calculations support the utility of this hybrid approximation.      

Sobolev-type approximation rates for divergence-free and curl-free RBF interpolants

Recently, error estimates have been made available for divergence-free radial basis function (RBF) interpolants. However, these results are only valid for functions within the associated reproducing kernel Hilbert space (RKHS) of the matrix-valued RBF. Functions within the associated RKHS, also known as the ``native space' of the RBF, can be characterized as vector fields having a specific smoothness, making the native space quite small. In this paper we develop Sobolev-type error estimates when the target function is less smooth than functions in the native space.

     

Sobolev Error Estimates and a Bernstein Inequality for Scattered Data Interpolation via Radial Basis Functions

Error estimates for scattered-data interpolation via radial basis functions (RBFs) for target functions in the associated reproducing kernel Hilbert space (RKHS) have been known for a long time. Recently, these estimates have been extended to apply to certain classes of target functions generating the data which are outside the associated RKHS. However, these classes of functions still were not "large" enough to be applicable to a number of practical situations. In this paper we obtain Sobolev-type error estimates on compact regions of Rⁿ when the RBFs have Fourier transforms that decay algebraically. In addition, we derive a Bernstein inequality for spaces of finite shifts of an RBF in terms of the minimal separation parameter.     

Approximation in rough native spaces by shifts of smooth kernels on spheres

Within the conventional framework of a native space structure, a smooth kernel generates a small native space, and “radial basis functions” stemming from the smooth kernel are intended to approximate only functions from this small native space. Therefore their approximation power is quite limited. Recently, Narcowich et al. (J. Approx. Theory 114 (2002) 70), and Narcowich and Ward (SIAM J. Math. Anal., to appear), respectively, have studied two approaches that have led to the empowerment of smooth radial basis functions in a larger native space. In the approach of [NW], the radial basis function interpolates the target function at some scattered (prescribed) points. In both approaches, approximation power of the smooth radial basis functions is achieved by utilizing spherical polynomials of a (possibly) large degree to form an intermediate approximation between the radial basis approximation and the target function. In this paper, we take a new approach. We embed the smooth radial basis functions in a larger native space generated by a less smooth kernel, and use them to approximate functions from the larger native space. Among other results, we characterize the best approximant with respect to the metric of the larger native space to be the radial basis function that interpolates the target function on a set of finite scattered points after the action of a certain multiplier operator. We also establish the error bounds between the best approximant and the target function.     

Error estimates of quasi-interpolation and its derivatives

Quasi-interpolation of radial basis functions on finite grids is a very useful strategy in approximation theory and its applications. A notable strongpoint of the strategy is to obtain directly the approximants without the need to solve any linear system of equations. For radial basis functions with Gaussian kernel, there have been more studies on the interpolation and quasi-interpolation on infinite grids. This paper investigates the approximation by quasi-interpolation operators with Gaussian kernel on the compact interval. The approximation errors for two classes of function with compact support sets are estimated. Furthermore, the approximation errors of derivatives of the approximants to the corresponding derivatives of the approximated functions are estimated. Finally, the numerical experiments are presented to confirm the accuracy of the approximations.     

Nonparametric lack-of-fit tests for parametric mean-regression models with censored data

We developed two kernel smoothing based tests of a parametric mean-regression model against a nonparametric alternative when the response variable is right-censored. The new test statistics are inspired by the synthetic data and the weighted least squares approaches for estimating the parameters of a (non)linear regression model under censoring. The asymptotic critical values of our tests are given by the quantiles of the standard normal law. The tests are consistent against fixed alternatives, local Pitman alternatives and uniformly over alternatives in Hölder classes of functions of known regularity.     

A Class of Large-Update and Small-Update Primal-Dual Interior-Point Algorithms for Linear Optimization

In this paper we present a class of polynomial primal-dual interior-point algorithms for linear optimization based on a new class of kernel functions. This class is fairly general and includes the classical logarithmic function, the prototype self-regular function, and non-self-regular kernel functions as special cases. The analysis of the algorithms in the paper follows the same line of arguments as in Bai et al. (SIAM J. Optim. 15:101–128, [2004]), where a variety of non-self-regular kernel functions were considered including the ones with linear and quadratic growth terms. However, the important case when the growth term is between linear and quadratic was not considered. The goal of this paper is to introduce such class of kernel functions and to show that the interior-point methods based on these functions have favorable complexity results. They match the currently best known iteration bounds for the prototype self-regular function with quadratic growth term, the simple non-self-regular function with linear growth term, and the classical logarithmic kernel function. In order to achieve these complexity results, several new arguments had to be used. This research is partially supported by the grant of National Science Foundation of China 10771133 and the Program of Shanghai Pujiang 06PJ14039.     

Closed form of the Fourier–Borel Kernel in the Framework of Clifford Analysis

In this paper we derive for the even dimensional case a closed form of the Fourier–Borel kernel in the Clifford analysis setting. This kernel is obtained as the monogenic component in the Fischer decomposition of the exponential function ${e^{\langle \underline{x}, \underline{u} \rangle}}$ where ${\langle . , . \rangle}$ denotes the standard inner product on the m-dimensional Euclidean space. A first approach based on Clifford analysis techniques leads to a conceptual formula containing the Gamma operator and the so-called Clifford–Bessel function, two fundamental objects in the theory of Clifford analysis. To obtain an explicit expression for the Fourier–Borel kernel in terms of a finite sum of Bessel functions, this formula remains however hard to work with. To that end we have also elaborated a more direct approach based on special functions leading to recurrence formulas for a closed form of the Fourier–Borel kernel.     

Fractional differential equations and Lyapunov functionals

We consider a scalar fractional differential equation, write it as an integral equation, and construct several Lyapunov functionals yielding qualitative results about the solution. It turns out that the kernel is convex with a singularity and it is also completely monotone, as is the resolvent kernel. While the kernel is not integrable, the resolvent kernel is positive and integrable with an integral value of one. These kernels give rise to essentially different types of Lyapunov functionals. It is to be stressed that the Lyapunov functionals are explicitly given in terms of known functions and they are differentiated using Leibniz’s rule. The results are readily accessible to anyone with a background of elementary calculus.     

Local uniform error estimates for spherical basis functions interpolation

This paper discusses local uniform error estimates for spherical basis functions (SBFs) interpolation, where error bounds for target functions are restricted on spherical cap. The discussion is first carried out in the native space associated with the smooth SBFs, which is generated by a strictly positive definite zonal kernel. Then, the smooth SBFs are embedded in a larger space that is generated by a less smooth kernel, and for the target functions outside the original native space, the local uniform error estimates are established. Finally, some numerical experiments are given to illustrate the theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

New parameterized kernel functions for linear optimization

Recent studies on the kernel function-based primal-dual interior-point algorithms indicate that a kernel function not only represents a measure of the distance between the iteration and the central path, but also plays a critical role in improving the computational complexity of an interior-point algorithm. In this paper, we propose a new class of parameterized kernel functions for the development of primal-dual interior-point algorithms for solving linear programming problems. The properties of the proposed kernel functions and corresponding parameters are investigated. The results lead to a complexity bounds of ${O\left(\sqrt{n}\,{\rm log}\,n\,{\rm log}\,\frac{n}{\epsilon}\right)}$ for the large-update primal-dual interior point methods. To the best of our knowledge, this is the best known bound achieved.     

Convex Kernel Underestimation of Functions with Multiple Local Minima

A function on Rⁿ with multiple local minima is approximated from below, via linear programming, by a linear combination of convex kernel functions using sample points from the given function. The resulting convex kernel underestimator is then minimized, using either a linear equation solver for a linear-quadratic kernel or by a Newton method for a Gaussian kernel, to obtain an approximation to a global minimum of the original function. Successive shrinking of the original search region to which this procedure is applied leads to fairly accurate estimates, within 0.0001% for a Gaussian kernel function, relative to global minima of synthetic nonconvex piecewise-quadratic functions for which the global minima are known exactly. Gaussian kernel underestimation improves by a factor of ten the relative error obtained using a piecewise-linear underestimator (O.L. Mangasarian, J.B. Rosen, and M.E. Thompson, Journal of Global Optimization, Volume 32, Number 1, Pages 1–9, 2005), while cutting computational time by an average factor of over 28.     

New derivation of the heisenberg kernel

The Heisenberg group gives rise to the simplest interesting example of a subellipticoperator. The hear kernel for this operator is known in terms of Fourier transforms. Here this hear kernel is derived in a simpleminded way from standard theroems in mathematical physics. A formula is given relating this kernel to the heat kernel of a magnetic field and ageneralization is given for similar geometries     

Kernel Logistic Regression and the Import Vector Machine

The support vector machine (SVM) is known for its good performance in two-class classification, but its extension to multiclass classification is still an ongoing research issue. In this article, we propose a new approach for classification, called the import vector machine (IVM), which is built on kernel logistic regression (KLR). We show that the IVM not only performs as well as the SVM in two-class classification, but also can naturally be generalized to the multiclass case. Furthermore, the IVM provides an estimate of the underlying probability. Similar to the support points of the SVM, the IVM model uses only a fraction of the training data to index kernel basis functions, typically a much smaller fraction than the SVM. This gives the IVM a potential computational advantage over the SVM.