Two-layer neural networks with values in a Banach space
This work addresses theoretical foundations for neural networks in infinite-dimensional spaces, which is incremental but relevant for researchers in functional analysis and machine learning theory.
The paper tackles the problem of approximating functions using two-layer neural networks with Banach space-valued inputs and outputs, extending finite-dimensional results to infinite-dimensional settings and proving inverse and direct approximation theorems with Monte-Carlo rates. It also studies optimal representation of such functions via signed measures from noisy observations, deriving convergence rates in a Bregman distance under source conditions.
We study two-layer neural networks whose domain and range are Banach spaces with separable preduals. In addition, we assume that the image space is equipped with a partial order, i.e. it is a Riesz space. As the nonlinearity we choose the lattice operation of taking the positive part; in case of $\mathbb R^d$-valued neural networks this corresponds to the ReLU activation function. We prove inverse and direct approximation theorems with Monte-Carlo rates for a certain class of functions, extending existing results for the finite-dimensional case. In the second part of the paper, we study, from the regularisation theory viewpoint, the problem of finding optimal representations of such functions via signed measures on a latent space from a finite number of noisy observations. We discuss regularity conditions known as source conditions and obtain convergence rates in a Bregman distance for the representing measure in the regime when both the noise level goes to zero and the number of samples goes to infinity at appropriate rates.