Measure theoretic results for approximation by neural networks with limited weights
This work addresses theoretical foundations for neural network approximation in mathematics, providing incremental insights into density conditions under constraints.
The paper tackles the problem of approximating continuous functions using single hidden layer neural networks with limited weight directions and thresholds, establishing a necessary and sufficient measure-theoretic condition for density and proving a density result for networks with a specific activation function and fixed neuron count.
In this paper, we study approximation properties of single hidden layer neural networks with weights varying on finitely many directions and thresholds from an open interval. We obtain a necessary and at the same time sufficient measure theoretic condition for density of such networks in the space of continuous functions. Further, we prove a density result for neural networks with a specifically constructed activation function and a fixed number of neurons.