Additive models in high dimensions

TL;DR

This paper explores the approximation of high-dimensional functions using additive models and ANOVA decompositions, demonstrating error bounds and illustrating their effectiveness through simulations.

Contribution

It provides theoretical error bounds for additive approximations in high dimensions and discusses conditions under which these bounds hold.

Findings

Error of order O(n^{m/2}) for ANOVA decompositions under smoothness conditions

Additive models effectively approximate high-dimensional functions

Simulations confirm theoretical error behavior

Abstract

We discuss some aspects of approximating functions on high-dimensional data sets with additive functions or ANOVA decompositions, that is, sums of functions depending on fewer variables each. It is seen that under appropriate smoothness conditions, the errors of the ANOVA decompositions are of order $O(n^{m/2})$ for approximations using sums of functions of up to $m$ variables under some mild restrictions on the (possibly dependent) predictor variables. Several simulated examples illustrate this behaviour.