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June 2000 Tensor product space ANOVA models
Yi Lin
Ann. Statist. 28(3): 734-755 (June 2000). DOI: 10.1214/aos/1015951996


To deal with the curse of dimensionality in high-dimensional nonparametric problems, we consider using tensor product space ANOVA models, which extend the popular additive models and are able to capture interactions of any order. The multivariate function is given an ANOVA decomposition, that is, it is expressed as a constant plus the sum of functions of one variable (main effects), plus the sum of functions of two variables (two-factor interactions)and so on. We assume the interactions to be in tensor product spaces.We show in both regression and white noise settings, the optimal rate of convergence for the TPS-ANOVA model is within a log factor of the one-dimensional optimal rate, and that the penalized likelihood estimator in TPS-ANOVA achieves this rate of convergence. The quick optimal rate of the TPS-ANOVA model makes it very preferable in high-dimensional function estimation. Many properties of the tensor product space of Sobolev –Hilbert spaces are also given.


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Yi Lin. "Tensor product space ANOVA models." Ann. Statist. 28 (3) 734 - 755, June 2000.


Published: June 2000
First available in Project Euclid: 12 March 2002

zbMATH: 1105.62329
MathSciNet: MR1792785
Digital Object Identifier: 10.1214/aos/1015951996

Primary: 62G07
Secondary: 62J20

Keywords: curse of dimensionality , Functional ANOVA , interaction , Optimal rate of convergence , penalized likelihood estimation , rate of convergence , smoothing splines , tensor product space , White noise model

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.28 • No. 3 • June 2000
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