On decompositions of multivariate functions

F. Y. Kuo, I. H. Sloan, G. W. Wasilkowski, H. WoźNiakowski

Research output: Contribution to journalArticlepeer-review

95 Scopus citations


We present formulas that allow us to decompose a function f of d variables into a sum of 2d terms fu indexed by subsets u of {1,..., d}, where each term fu depends only on the variables with indices in u. The decomposition depends on the choice of d commuting projections {Pj}dj=1where Pj (f) does not depend on the variable xj. We present an explicit formula for fu, 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 z, then, for every choice of {Pj}dj=1 the terms fu = 0 for all subsets u containing z. Furthermore, in a reproducing kernel Hilbert space setting, we give sufficient conditions for the terms fu to be mutually orthogonal.

Original languageEnglish
Pages (from-to)953-966
Number of pages14
JournalMathematics of Computation
Issue number270
StatePublished - Apr 2010

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics


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