Piecewise polynomial kernel networks

We introduce a network structure for function approximation based on a combination of kernels with compact support. The kernel functions that we use are continuous piecewise polynomials (CPP) defined over a rectangular lattice. For the representation of CPP's the method of B-splines is used when kernels of the most smooth kind are required. In the paper, we also introduce the localized threshold decomposition operator, as a method to use when the least smooth kernels are required for the approximation. Both methods can be combined for improved results. CPP kernel networks are shape adaptive since the kernel shapes can be fitted to match the local characteristics of the function being approximated. Moreover, the shape parameters are linear and therefore they can be identified using fast linear estimation procedures. For the estimation of kernel locations we use a successive approximation learning algorithm. In this paper, a channel equalization example is used to validate the concepts introduced. We show that the shape adaptation property, in this case, allows the networks to improve by a large margin the results of other standard equalization methods.