Fast high-dimensional approximation with sparse occupancy trees

This paper is concerned with scattered data approximation in high dimensions: Given a data set X ⊂ Rd of N data points xi along with values yi ∈ Rd , i = 1, . . . , N , and viewing the yi as values yi = f(xi) of some unknown function f , we wish to return for any query point x ∈ Rd an approximation f̃(x) to y = f(x). Here the spatial dimension d should be thought of as large. We wish to emphasize that we do not seek a representation of f̃ in terms of a fixed set of trial functions but define f̃ through recovery schemes which, in the first place, are designed to be fast and to deal efficiently with large data sets. For this purpose we propose new methods based on what we call sparse occupancy trees and partitioning schemes based on simplex subdivisions.