Approximation of functions by neural networks
This provides a theoretical foundation for neural network approximation in machine learning, addressing a fundamental problem for researchers in approximation theory and AI.
The paper tackles the problem of approximating measurable functions on the hypercube using affine neural networks, achieving an approximation up to any precision ε with a bounded number of neurons that depends only on ε, not on the function or dimension n.
We study the approximation of measurable functions on the hypercube by functions arising from affine neural networks. Our main achievement is an approximation of any measurable function $f \colon W_n \to [-1,1]$ up to a prescribed precision $\varepsilon>0$ by a bounded number of neurons, depending only on $\varepsilon$ and not on the function $f$ or $n \in \mathbb N$.