Fast high-dimensional approximation with sparse occupancy trees

Abstract

This paper is concerned with scattered data approximation in high dimensions: Given a data set X ⊂ R d of N data points x i along with values y i ∈ R d ′ , i = 1 , … , N , and viewing the y i as values y i = f ( x i ) of some unknown function f , we wish to return for any query point x ∈ R d an approximation f ̃ ( x ) to y = f ( x ) . Here the spatial dimension d should be thought of as large. We emphasize that we do not seek a representation of f ̃ in terms of a fixed set of trial functions but define f ̃ through recovery schemes which are primarily 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 piecewise linear schemes based on simplex subdivisions.