  Estimation of a conditional mean (linking a set of features to an outcome of
interest) is a fundamental statistical task. While there is an appeal to
flexible nonparametric procedures, effective estimation in many classical
nonparametric function spaces (e.g., multivariate Sobolev spaces) can be
prohibitively difficult -- both statistically and computationally -- especially
when the number of features is large. In this paper, we present (penalized)
sieve estimators for regression in nonparametric tensor product spaces: These
spaces are more amenable to multivariate regression, and allow us to, in-part,
avoid the curse of dimensionality. Our estimators can be easily applied to
multivariate nonparametric problems and have appealing statistical and
computational properties. Moreover, they can effectively leverage additional
structures such as feature sparsity. In this manuscript, we give theoretical
guarantees, indicating that the predictive performance of our estimators scale
favorably in dimension. In addition, we also present numerical examples to
compare the finite-sample performance of the proposed estimators with several
popular machine learning methods.
