On decompositions of multivariate functions

Abstract:

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