Sequential function approximation with noisy data

Highlights: • We proposed a method for approximating unknown target function using large sample sets with noises. • We proposed a mathematical framework for sequential approximation (SA) method, which allows us to define the method in a unified manner. • We extend the analysis of the SA method to noisy data case, which was not considered by the previous work. • We provided extensive numerical examples to demonstrate the performance of the method. We present a sequential method for approximating an unknown function sequentially using random noisy samples. Unlike the traditional function approximation methods, the current method constructs the approximation using one sample at a time. This results in a simple numerical implementation using only vector operations and avoids the need to store the entire data set. The method is thus particularly suitable when data set is exceedingly large. Furthermore, we present a general theoretical framework to define and interpret the method. Both upper and lower bounds of the method are established for the expectation of the results. Numerical examples are provided to verify the theoretical findings.