Abstract. We explore a connection between Gaussian radial basis functions and polynomials. Using standard tools of potential theory, we find that these radial functions are susceptible to the Runge phenomenon, not only in the limit of increasingly flat functions, but also in the finite shape parameter case. We show that there exist interpolation node distributions that prevent such phenomena an...

Several radial basis function based methods contain a free shape parameter which has  a crucial role in the accuracy of the methods. Performance evaluation of this parameter in different  functions with various data has always been a topic of study. In the present paper, we consider studying the methods which determine an optimal value for the shape parameter in interpolations of radial basis  ...

In this study, the radial basis functions based SG algorithm (SGRBF) is applied for evolution of level sets in image segmentation. The implementation of level set method in image processing often involves solving partial differential equations (PDEs). Finite differences implicit scheme is a prevalent method to solve PDE for extending the evolution of level sets. Instead of using finite differen...

Conventionally, in radial basis function (RBF) network width factor is constructed by obtaining r-nearest neighbor rule or taking equal to a constant for all Gaussian functions. This paper proposes an approach for the construction of width factor using genetic algorithm to optimize the Gaussian function. Our experimental results show that our proposed optimal-based width outperforms the convent...

It is well-known that the classical univariate orthogonal polynomials give rise to highly efficient Gaussian quadrature rules. We show how these classical families of polynomials can be generalized to a multivariate setting and how this generalization leads to truly Gaussian cubature rules for specific families of multivariate polynomials. The multivariate homogeneous orthogonal functions that ...

We present a practical way of introducing convolutional structure into Gaussian processes, making them more suited to high-dimensional inputs like images. The main contribution of our work is the construction of an inter-domain inducing point approximation that is well-tailored to the convolutional kernel. This allows us to gain the generalisation benefit of a convolutional kernel, together wit...

We present a fast direct algorithm for computing symmetric factorizations, i.e. A = WWT , of symmetric positive-definite hierarchical matrices with weak-admissibility conditions. The computational cost for the symmetric factorization scales as O(n log n) for hierarchically off-diagonal low-rank matrices. Once this factorization is obtained, the cost for inversion, application, and determinant c...

The efficiency of the orthogonal least squares (OLS) method for training approximation networks is examined using the criterion of energy compaction. We show that the selection of basis vectors produced by the procedure is not the most compact when the approximation is performed using a nonorthogonal basis. Hence, the algorithm does not produce the smallest possible networks for a given approxi...