Fredholm Integral Equations for Function Approximation and the Training of Neural Networks

We present a novel and mathematically transparent approach to function approximation and the training of large, high-dimensional neural networks, based on the approximate least-squares solution of associated Fredholm integral equations of the first kind by Ritz-Galerkin discretization, Tikhonov regularization, and tensor train methods. Practical application to supervised learning problems of regression and classification type confirm that the resulting algorithms are competitive with stateof-the-art neural network--based methods.