Proofs confirm strong universality of single-neuron models over fields
Proofs of strong forms of the Bovdi--Geche universality conjecture and a complex analogue
Machine Learning
Summary
Understanding how simple neural networks can represent many functions is important for AI and math. This paper proves that even a single neuron, with special mathematical activation functions, can approximate a wide range of functions over finite fields and complex numbers. The authors answered long-standing questions by using advanced math tools about number theory and geometry. These results extend our understanding of neural network capabilities in very abstract settings.
What this means in practice
- •For cryptography engineers: Design cryptographic algorithms that use finite field neural models with guaranteed function approximation properties.
- •For quantum computing developers: Explore encoding complex function approximations in neural models relevant to quantum state representations using complex neurons.
A theory result. No direct application yet.
Authors
Victor Bovdi, Ambrus Pal
Abstract
We prove two universality theorems for neural networks consisting of a single neuron. First we prove a strong form of the Bovdi--Geche universality conjecture on neurons over finite fields, where the activation function is a certain modification of the discrete logarithm, using the Weil bound on multiplicative character sums over finite fields. Then we show a generalisation of Aizenberg's universality theorem for complex binary neurons with a weight vector independent of the target function, positively answering another question in the literature. The proof of the latter uses an equidistribution theorem for the dilates of certain maps into tori.