Neural Networks With Dense Weights Are Not Universal Approximators

TL;DR

This paper demonstrates that dense neural networks with constrained weights cannot universally approximate all continuous functions, revealing fundamental limitations and emphasizing the importance of sparse connectivity.

Contribution

It provides a theoretical proof that dense neural networks under natural constraints lack universality, challenging assumptions from classical approximation theorems.

Findings

Dense neural networks cannot approximate certain Lipschitz functions.

Universal approximation requires sparse connectivity under weight constraints.

The results highlight intrinsic limitations of dense architectures.

Abstract

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.