Simplicial Neural Networks:First Steps and Future Applications

Anonymous

Keywords: Universal Approximation Theorem, Simplicial Approximation Theorem, Multi-layer feed-forward network, Triangulations

TL;DR: A constructive approach to the Universal Approximation Theorem based on the Simplicial Approximation Theorem.

Abstract:

It is well-known that artificial neural networks are universal approximators. The Universal Approximation Theorem
proves that, given a continuous function on a compact set embedded in an $n$-dimensional space, there exists a one-hidden-layer feed-forward network that approximates the function; however, it does not provide a way of building such a network. In
previous work, the authors presented a constructive approach to tackle this problem for the case of a continuous function on triangulated spaces by connecting the Simplicial Approximation Theorem, a classical result from algebraic topology, and the Universal Approximation Theorem. In this paper, we revisit such a result and propose future applications.

Previous Submission: Yes