Directly Controlling the Sparsity of an Interpolation Matrix Formed by Compactly Supported RBFNs

Abstract

When applying radial basis function networks (RBFNs) for interpolating a function with a large number of nodes in near real-time, one must consider three potential limitations imposed by a computer system: 1) storage limitations, 2) computation time limitations, and 3) solution stability limitations. By using compactly supported radial basis functions (CSRBFs), one can alleviate these limitations. Using CSRBFs leads to a sparse interpolation matrix, which allows special techniques to be used to reduce the memory required to store the interpolation matrix and to reduce the computational complexity required to solve the interpolation problem. They also lead to better conditioned interpolation matrices due to their localness. However, it has been shown in the literature that although highly local radial basis functions lead to better conditioned interpolation matrices, they typically produce poor interpolation results. In this paper, we introduce an algorithm for directly controlling the degree of sparsity of an interpolation matrix formed by CSRBFs with data nodes on a uniform grid. Given a desired degree of sparsity, the algorithm determines the support radius required for CSRBFs to produce an interpolation matrix with the desired degree of sparsity. By being able to directly specify the degree of sparsity of the interpolation matrix, a practitioner can more intuitively balance memory requirements, computation time requirements, and the stability and accuracy of the interpolation results.