Abstract
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 language | English |
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Pages (from-to) | 953-966 |
Number of pages | 14 |
Journal | Mathematics of Computation |
Volume | 79 |
Issue number | 270 |
DOIs | |
State | Published - Apr 2010 |
ASJC Scopus subject areas
- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics