Hyperpolation

post by Gunnar_Zarncke · 2024-09-15T21:37:00.002Z · LW · GW · 6 comments

This is a link post for https://arxiv.org/pdf/2409.05513

Contents

6 comments

Interpolation, Extrapolation, Hyperpolation: Generalising into new dimensions

by Toby Ord 

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.

And it turns out that it is as doable systematically as extrapolation. For example, before reading the paper, can you guess an f(x,y) that is a simple reasonable generalization of this f(x)?

6 comments

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comment by niplav · 2024-09-16T18:45:13.403Z · LW(p) · GW(p)

I'm surprised that the paper doesn't mention analytic continuations of complex functions—maybe that is also taken as an instance of extrapolation?

Replies from: Gunnar_Zarncke
↑ comment by Gunnar_Zarncke · 2024-09-16T22:23:40.222Z · LW(p) · GW(p)

I think it would have been worth mentioning! Analytics continuation always stays in the original manifold and thus doesn't extend into another dimension. It is true, though, that the local extension can lead to Riemann surfaces that can eventually "wrap around" and form "layers" that have multiple different value at the same original coordinate. This can be interpreted as pointing into another dimension (even though it is not doing so continuously (real-values)). I would guess there is a subset of methods for Hyperpolation that use this approach to hyperpolate some functions (maybe for functions for which analytical continuation isn't known yet).

comment by cousin_it · 2024-09-16T14:06:12.757Z · LW(p) · GW(p)

I'm not sure the article fully justifies the thesis. It shows hyperpolation only for a handful of cases where the given function is a slice of a simpler higher-dimensional function. But interpolation isn't limited to such cases. Interpolation is general: given any set of (x,y) pairs, there are reasonable ways to interpolate between them. Is there a nontrivial way of doing hyperpolation in general?

Replies from: Gunnar_Zarncke
↑ comment by Gunnar_Zarncke · 2024-09-16T22:34:22.132Z · LW(p) · GW(p)

The article gives a general but boring example: Extrusion. This is analogous to stepwise interpolation ( such that ), which is also boring. Depending on the constraints put on the function different methods result. For example, analytical functions can plausibly extended in the complex plane by choosing the simplest coefficients.

comment by LeahBryant · 2024-10-04T11:48:51.415Z · LW(p) · GW(p)

Yes, you are right, I agree with you.

Replies from: Gunnar_Zarncke
↑ comment by Gunnar_Zarncke · 2024-10-05T15:01:16.824Z · LW(p) · GW(p)

Hi Newbie, what are your thoughts on it?