Incremental Radial Basis Function Computation for Neural Networks

Abstract

This paper present a novel approach for incremental computation of Radial Basis Functions (RBF) for Fuzzy Systems and Neural Networks with computational complexity of O(N2) is presented. This technique enables efficient insertion of new data and removal of selected or invalid data. RBF are used across many fields, including geometrical, image processing and pattern recognition, medical applications, signal processing, speech recognition, etc. The main prohibitive factor is the computational cost of the RBF computation for larger data sets or if data set is changed and RBFs have to be recomputed. The presented technique is applicable in general to fuzzy systems as well offering a significant speed up due to lower computational complexity of the presented approach. The Incremental RBF Computation enables also fast RBF recomputation on “sliding window” data due to fast insert/remove operations. This is a very significant factor especially in guided Neural Networks case. Generally, interpolation based on RBF is very often used for scattered scalar data interpolation in n-dimensional space. As there is no explicit order in data sets, computations are quite time consuming that leads to limitation of usability even for static data sets. Computational complexity of RBF for N values is of O(N3) or O(k N2), k is a number of iterations if an iterative method is used, which is prohibitive for many real applications. The inverse matrix can also be computed by the Strassen algorithm based on matrix block notation with O(N2.807) complexity. Even worst situation occurs when interpolation has to be made over non-constant data sets, as the whole set of equations for determining RBFs has to be recomputed when data set is changed. This situation is typical for applications in which some values are becoming invalid and new values are acquired.