Approximation with Random Bases: Pro et Contra
This work provides theoretical insights into the limitations of random basis approximation for practitioners using neural networks in modeling and control.
The paper analyzes randomized and deterministic procedures for selecting approximators from dense families of parameterized functions, showing that both achieve O(1/N) convergence in L2 norm only when additional information is provided; without it, exponential growth or extreme parameter sensitivity can occur, with implications for neural networks.
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.