On decompositions of multivariate functions

Abstract

We present formulas that allow us to decompose a function  f f of  d d variables into a sum of  2 d 2^d terms f u f_{\mathbf {u}} indexed by subsets u \mathbf {u} of { 1 , … , d } \{1,\ldots ,d\} , where each term f u f_{\mathbf {u}} depends only on the variables with indices in u \mathbf {u} . The decomposition depends on the choice of d d commuting projections { P j } j = 1 d \{P_j\}_{j=1}^d , where P j ( f ) P_j(f) does not depend on the variable x j x_j . We present an explicit formula for f u f_{\mathbf {u}} , which is new even for the anova and anchored decompositions; both are special cases of the general decomposition. We show that the decomposition is minimal in the following sense: if f f is expressible as a sum in which there is no term that depends on all of the variables indexed by the subset z \mathbf {z} , then, for every choice of { P j } j = 1 d \{P_j\}_{j=1}^d , the terms f u = 0 f_{\mathbf {u}}=0 for all subsets u \mathbf {u} containing  z \mathbf {z} . Furthermore, in a reproducing kernel Hilbert space setting, we give sufficient conditions for the terms f u f_{\mathbf {u}} to be mutually orthogonal.