Optimal sampling points in reproducing kernel Hilbert spaces

The recent development of compressed sensing seeks to extract information from as few samples as possible. In such applications, since the number of samples is restricted, one should deploy the sampling points wisely. We are motivated to study the optimal distribution of finite sampling points in reproducing kernel Hilbert spaces, the natural background function spaces for sampling. Formulation under the framework of optimal reconstruction yields a minimization problem. In the discrete measure case, we estimate the distance between the optimal subspace resulting from a general Karhunen–Loève transform and the kernel space to obtain another algorithm that is computationally favorable. Numerical experiments are then presented to illustrate the effectiveness of the algorithms for the searching of optimal sampling points.     

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.     

Weighted reproducing kernels and Toeplitz operators on harmonic Bergman spaces on the real ball

For the standard weighted Bergman spaces on the complex unit ball, the Berezin transform of a bounded continuous function tends to this function pointwise as the weight parameter tends to infinity. We show that this remains valid also in the context of harmonic Bergman spaces on the real unit ball of any dimension. This generalizes the recent result of C. Liu for the unit disc, as well as the original assertion concerning the holomorphic case. Along the way, we also obtain a formula for the corresponding weighted harmonic Bergman kernels.

     

Average sampling and reconstruction for reproducing kernel stochastic signals

This paper mainly considers the problem of reconstructing a reproducing kernel stochastic signal from its average samples. First, a uniform convergence result for reconstructing the deterministic reproducing kernel signals by an iterative algorithm is established. Then, we prove that the quadratic sum of the corresponding reconstructed functions is uniformly bounded. Moreover, the reconstructed functions provide a frame expansion in the special case p = 2. Finally, the mean square convergence for recovering a weighted reproducing kernel stochastic signal from its average samples is given under some decay condition for the autocorrelation function, which can be removed for the case p = 2. Copyright © 2015 John Wiley & Sons, Ltd.     

Orthogonality from disjoint support in reproducing kernel Hilbert spaces

We investigate reproducing kernel Hilbert spaces (RKHS) where two functions are orthogonal whenever they have disjoint support. Necessary and sufficient conditions in terms of feature maps for the reproducing kernel are established. We also present concrete examples of finite dimensional RKHS and RKHS with a translation invariant reproducing kernel. In particular, it is shown that a Sobolev space has the orthogonality from disjoint support property if and only if it is of integer index.     

Bounded Toeplitz products on the weighted Bergman spaces of the unit ball

Let p>1 and let q denote the number such that (1/p)+(1/q)=1. We give a necessary condition for the product of Toeplitz operators to be bounded on the weighted Bergman space of the unit ball (α>−1), where and , as well as a sufficient condition for to be bounded on . We use techniques different from those in [K. Stroethoff, D. Zheng, Bounded Toeplitz products on Bergman spaces of the unit ball, J. Math. Anal. Appl. 325 (2007) 114-129], in which the case p=2 was proved.     

Continuum versus discrete networks,graph Laplacians,and reproducing kernel Hilbert spaces

Motivated by applications to machine learning, we construct a reversible and irreducible Markov chain whose state space is a certain collection of measurable sets of a chosen l.c.h. space $\mathfrak{X}$. We study the resulting network (connected undirected graph), including transience, Royden and Riesz decompositions, and kernel factorization. We describe a construction for Hilbert spaces of signed measures which comes equipped with a new notion of reproducing kernels and there is a unique solution to a regularized optimization problem involving the approximation of $L^2$ functions by functions of finite energy. The latter has applications to machine learning (for Markov random fields, for example).     

Quantile regression with an epsilon-insensitive loss in a reproducing kernel Hilbert space

This paper proposes a method to estimate the conditional quantile function using an epsilon-insensitive loss in a reproducing kernel Hilbert space. When choosing a smoothing parameter in nonparametric frameworks, it is necessary to evaluate the complexity of the model. In this regard, we provide a simple formula for computing an effective number of parameters when implementing an epsilon-insensitive loss. We also investigate the effects of the epsilon-insensitive loss.     

Approximate solution of the Fredholm integral equation of the first kind in a reproducing kernel Hilbert space

An approach for solving Fredholm integral equations of the first kind is proposed for in a reproducing kernel Hilbert space (RKHS). The interest in this problem is strongly motivated by applications to actual prospecting. In many applications one is puzzled by an ill-posed problem in space C[a,b] or L²[a,b], namely, measurements of the experimental data can result in unbounded errors of solutions of the equation. In this work, the representation of solutions for Fredholm integral equations of the first kind is obtained if there are solutions and the stability of solutions is discussed in RKHS. At the same time, a conclusion is obtained that approximate solutions are also stable with respect to _∞ or _L² in RKHS. A numerical experiment shows that the method given in the work is valid.     

Stochastic processes with sample paths in reproducing kernel Hilbert spaces

A theorem of M. F. Driscoll says that, under certain restrictions, the probability that a given Gaussian process has its sample paths almost surely in a given reproducing kernel Hilbert space (RKHS) is either or . Driscoll also found a necessary and sufficient condition for that probability to be .

Doing away with Driscoll's restrictions, R. Fortet generalized his condition and named it nuclear dominance. He stated a theorem claiming nuclear dominance to be necessary and sufficient for the existence of a process (not necessarily Gaussian) having its sample paths in a given RKHS. This theorem - specifically the necessity of the condition - turns out to be incorrect, as we will show via counterexamples. On the other hand, a weaker sufficient condition is available.

Using Fortet's tools along with some new ones, we correct Fortet's theorem and then find the generalization of Driscoll's result. The key idea is that of a random element in a RKHS whose values are sample paths of a stochastic process. As in Fortet's work, we make almost no assumptions about the reproducing kernels we use, and we demonstrate the extent to which one may dispense with the Gaussian assumption.

     

Construction and calculation of reproducing kernel determined by various linear differential operators

We presented a method to construct and calculate the reproducing kernel for the linear differential operator with constant coefficients and a single latent root; further, we gave the formula for calculation. Additionally, by studying the recurrence relation of the reproducing kernel with arithmetic latent roots, we found that the reproducing kernel with a multi-knots interpolation constraint can be concisely represented by one with an initial-value constraint.     

Optimal rates for spectral algorithms with least-squares regression over Hilbert spaces

In this paper, we study regression problems over a separable Hilbert space with the square loss, covering non-parametric regression over a reproducing kernel Hilbert space. We investigate a class of spectral/regularized algorithms, including ridge regression, principal component regression, and gradient methods. We prove optimal, high-probability convergence results in terms of variants of norms for the studied algorithms, considering a capacity assumption on the hypothesis space and a general source condition on the target function. Consequently, we obtain almost sure convergence results with optimal rates. Our results improve and generalize previous results, filling a theoretical gap for the non-attainable cases.     

Unconditional reproducing kernel bases in de Branges spaces

We describe unconditional bases of the form {k(z,λ _n ):λ _n ∈Λ}, Λ∩?=? in de Branges spaces, where k is the reproducing kernel.     

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.     

Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method

This paper investigates the numerical solutions of singular second order three-point boundary value problems using reproducing kernel Hilbert space method. It is a relatively new analytical technique. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel Hilbert space method cannot be used directly to solve a singular second order three-point boundary value problem, so we convert it into an equivalent integro-differential equation, which can be solved using reproducing kernel Hilbert space method. Four numerical examples are given to demonstrate the efficiency of the present method. The numerical results demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.     

Average sampling and reconstruction in a reproducing kernel subspace of homogeneous type space

In this paper, we first introduce a reproducing kernel subspace of , where is a homogeneous type space. Then we consider average sampling and reconstruction of signals in the reproducing kernel subspace of . We show that signals in the reproducing kernel subspace of could be stably reconstructed from its average samples taken on a relatively‐separated set with small gap. Exponential convergence is established for the iterative approximation‐projection reconstruction algorithm.     

Application of reproducing kernel method to third order three-point boundary value problems

This paper investigates the analytical approximate solutions of third order three-point boundary value problems using reproducing kernel method. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel method can not be used directly to solve third order three-point boundary value problems, since there is no method of obtaining reproducing kernel satisfying three-point boundary conditions. This paper presents a method for solving reproducing kernel satisfying three-point boundary conditions so that reproducing kernel method can be used to solve third order three-point boundary value problems. Results of numerical examples demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.