This is a linkpost for https://arxiv.org/pdf/2409.05513
I'm surprised that the paper doesn't mention analytic continuations of complex functions—maybe that is also taken as an instance of extrapolation?
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).
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?
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.
Interpolation, Extrapolation, Hyperpolation: Generalising into new dimensions
by Toby Ord
Abstract:
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)?