Why do networks have inhibitory/negative connections?

TL;DR

This paper demonstrates that negative weights are essential for neural networks to be universal function approximators, providing a formal theoretical foundation for the biological and artificial necessity of inhibitory connections.

Contribution

It offers the first formal proof that non-negative neural networks cannot be universal approximators, highlighting the importance of negative weights for representation capacity.

Findings

Non-negative neural networks are not universal approximators.

Negative weights enable richer function representations.

Insights into geometric properties of the representation space.

Abstract

Why do brains have inhibitory connections? Why do deep networks have negative weights? We propose an answer from the perspective of representation capacity. We believe representing functions is the primary role of both (i) the brain in natural intelligence, and (ii) deep networks in artificial intelligence. Our answer to why there are inhibitory/negative weights is: to learn more functions. We prove that, in the absence of negative weights, neural networks with non-decreasing activation functions are not universal approximators. While this may be an intuitive result to some, to the best of our knowledge, there is no formal theory, in either machine learning or neuroscience, that demonstrates why negative weights are crucial in the context of representation capacity. Further, we provide insights on the geometric properties of the representation space that non-negative deep networks cannot represent. We expect these insights will yield a deeper understanding of more sophisticated inductive priors imposed on the distribution of weights that lead to more efficient biological and machine learning.