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.