The Annals of Statistics

Tensor product space ANOVA models

Yi Lin

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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.

Article information

Ann. Statist., Volume 28, Number 3 (2000), 734-755.

First available in Project Euclid: 12 March 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation
Secondary: 62J20: Diagnostics

Functional ANOVA tensor product space white noise model rate of convergence optimal rate of convergence penalized likelihood estimation interaction curse of dimensionality smoothing splines


Lin, Yi. Tensor product space ANOVA models. Ann. Statist. 28 (2000), no. 3, 734--755. doi:10.1214/aos/1015951996.

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