Approximation with Random Bases: Pro et Contra

TL;DR

This paper analyzes methods for selecting parameterized functions for approximation, highlighting the importance of additional information for success and discussing implications for neural network applications.

Contribution

It provides a comparative analysis of randomized and deterministic approximation procedures, emphasizing the role of prior information in ensuring convergence.

Findings

Success of approximation methods depends on additional information about function families.

Without extra information, the number of terms needed can grow exponentially.

Approximation sensitivity increases without proper prior knowledge.

Abstract

In this work we discuss the problem of selecting suitable approximators from families of parameterized elementary functions that are known to be dense in a Hilbert space of functions. We consider and analyze published procedures, both randomized and deterministic, for selecting elements from these families that have been shown to ensure the rate of convergence in $L_2$ norm of order $O(1/N)$, where $N$ is the number of elements. We show that both randomized and deterministic procedures are successful if additional information about the families of functions to be approximated is provided. In the absence of such additional information one may observe exponential growth of the number of terms needed to approximate the function and/or extreme sensitivity of the outcome of the approximation to parameters. Implications of our analysis for applications of neural networks in modeling and control are illustrated with examples.