TY - JOUR

T1 - Random vector functional link networks for function approximation on manifolds

AU - Needell, D

AU - Nelson, AA

AU - Saab, R

AU - Salanevich, P

AU - Schavemaker, O

N1 - Publisher Copyright:
Copyright © 2024 Needell, Nelson, Saab, Salanevich and Schavemaker.

PY - 2024/4/17

Y1 - 2024/4/17

N2 - 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 study, we begin to fill this theoretical gap. We then extend this result to the non-asymptotic setting using a concentration inequality for Monte-Carlo integral approximations. 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 squared decaying asymptotically like O(1/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.

AB - 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 study, we begin to fill this theoretical gap. We then extend this result to the non-asymptotic setting using a concentration inequality for Monte-Carlo integral approximations. 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 squared decaying asymptotically like O(1/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.

KW - feed-forward neural networks

KW - function approximation

KW - machine learning

KW - random vector functional link

KW - smooth manifold

UR - https://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=d7dz6a2i7wiom976oc9ff2iqvdhv8k5x&SrcAuth=WosAPI&KeyUT=WOS:001210857900001&DestLinkType=FullRecord&DestApp=WOS_CPL

UR - http://www.scopus.com/inward/record.url?scp=85191789120&partnerID=8YFLogxK

U2 - 10.3389/fams.2024.1284706

DO - 10.3389/fams.2024.1284706

M3 - Article

SN - 2297-4687

VL - 10

JO - Frontiers in Applied Mathematics and Statistics

JF - Frontiers in Applied Mathematics and Statistics

M1 - 1284706

ER -