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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On decompositions of multivariate functions


Authors: F. Y. Kuo, I. H. Sloan, G. W. Wasilkowski and H. Wozniakowski
Journal: Math. Comp. 79 (2010), 953-966
MSC (2000): Primary 41A63, 41A99
Published electronically: November 20, 2009
MathSciNet review: 2600550
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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.


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Additional Information

F. Y. Kuo
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia
Email: f.kuo@unsw.edu.au

I. H. Sloan
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia
Email: i.sloan@unsw.edu.au

G. W. Wasilkowski
Affiliation: Department of Computer Science, University of Kentucky, Lexington, Kentucky 40506
Email: greg@cs.uky.edu

H. Wozniakowski
Affiliation: Department of Computer Science, Columbia University, New York, New York 10027, and Institute of Applied Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland
Email: henryk@cs.columbia.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-09-02319-9
PII: S 0025-5718(09)02319-9
Received by editor(s): May 2, 2008
Received by editor(s) in revised form: October 16, 2008, and February 12, 2009
Published electronically: November 20, 2009
Article copyright: © Copyright 2009 American Mathematical Society