How to interpolate multidimensional functions?

Question

I'm learning about interpolation and I wanted to ask if there's a "good" method to interpolate multidimensional functions (when the dimension can be even a few thousands)?

Is there a theoretic limit on interpolating such functions?

Thanks

Answer 1

While any mesh-, grid-, tile- based methods become generally impractical in high dimensions due to the exponentially growing number of nodes, interpolation with radial basis functions would suffer only due to growing function sampling and/or increasing the support radius in case of compactly supported RBF. High-dimensional space is very sparse, that's the "curse of dimensionality". Methods like data clustering, projecting, other methods from statistics and data mining are used to somehow reduce the problem dimensionality first.

Theoretically, yes, one can write the RBF interpolation problem for high dimensions and it even looks the same but it would grow impossible to solve for the sheer amount of points in high dimensions.