Dense Neural Networks are not Universal Approximators
This work identifies a fundamental limitation of dense neural networks for approximation theory, motivating the necessity of sparse connectivity for universal approximation.
The paper proves that dense neural networks with natural weight constraints are not universal approximators, showing there exist Lipschitz continuous functions they cannot approximate, which challenges the common belief in their universality.
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. We consider ReLU neural networks subject to natural constraints on weights and input and output dimensions, which model a notion of dense connectivity. Within this setting, we demonstrate the existence of Lipschitz continuous functions that cannot be approximated by such networks. This highlights intrinsic limitations of neural networks with dense layers and motivates the use of sparse connectivity as a necessary ingredient for achieving true universality.