Interpolation, Extrapolation, Hyperpolation: Generalising into new dimensions

TL;DR

This paper introduces hyperpolation, a new method for estimating function values outside existing data subspaces, linking it to creativity and AI limitations.

Contribution

It proposes hyperpolation as a novel generalization technique beyond interpolation and extrapolation, exploring its implications for creativity and AI.

Findings

Hyperpolation enables estimation outside data subspaces.

Hyperpolation relates to creativity in arts and sciences.

Current AI systems lack hyperpolation capabilities.

Abstract

This paper introduces the concept of hyperpolation: a way of generalising from a limited set of data points that is a peer to the more familiar concepts of interpolation and extrapolation. Hyperpolation is the task of estimating the value of a function at new locations that lie outside the subspace (or manifold) of the existing data. We shall see that hyperpolation is possible and explore its links to creativity in the arts and sciences. We will also examine the role of hyperpolation in machine learning and suggest that the lack of fundamental creativity in current AI systems is deeply connected to their limited ability to hyperpolate.