Dense Neural Networks are not Universal Approximators

ArXi:2602.07618v3 Announce Type: replace We investigate the approximation capabilities of dense neural networks. While universal approximation theorems establish that sufficiently large architectures can approximate arbitrary continuous functions if there are no restrictions on the weight values, we show that dense neural networks do not possess this universality. Our argument is based on a model compression approach, combining the weak regularity lemma with an interpretation of feedforward networks as message passing graph neural networks.