Random Vector Functional Link Networks for Function Approximation on Manifolds

TL;DR

This paper provides a rigorous theoretical foundation for random vector functional link networks, demonstrating their universal approximation capabilities on manifolds and establishing error bounds with practical implications.

Contribution

It offers the first rigorous proof of the universal approximation property of RVFL networks on manifolds, including asymptotic and non-asymptotic error bounds, extending prior empirical success.

Findings

RVFL networks are universal approximators for continuous functions on compact domains.

Approximation error decays asymptotically like O(1/√n) with the number of nodes.

Theoretical guarantees are extended to functions on smooth, compact submanifolds.

Abstract

The learning speed of feed-forward neural networks is notoriously slow and has presented a bottleneck in deep learning applications for several decades. For instance, gradient-based learning algorithms, which are used extensively to train neural networks, tend to work slowly when all of the network parameters must be iteratively tuned. To counter this, both researchers and practitioners have tried introducing randomness to reduce the learning requirement. Based on the original construction of Igelnik and Pao, single layer neural-networks with random input-to-hidden layer weights and biases have seen success in practice, but the necessary theoretical justification is lacking. In this paper, we begin to fill this theoretical gap. We provide a (corrected) rigorous proof that the Igelnik and Pao construction is a universal approximator for continuous functions on compact domains, with approximation error decaying asymptotically like $O(1/\sqrt{n})$ for the number $n$ of network nodes. We then extend this result to the non-asymptotic setting, proving that one can achieve any desired approximation error with high probability provided $n$ is sufficiently large. We further adapt this randomized neural network architecture to approximate functions on smooth, compact submanifolds of Euclidean space, providing theoretical guarantees in both the asymptotic and non-asymptotic forms. Finally, we illustrate our results on manifolds with numerical experiments.