Regression in tensor product spaces by the method of sieves

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 some sieve estimators for regression in multivariate product spaces. We take Sobolev-type smoothness spaces as an example, though our general framework can be applied to many reproducing kernel Hilbert spaces. These spaces are more amenable to multivariate regression, and allow us to, inpart, avoid the curse of dimensionality. Our estimator can be easily applied to multivariate nonparametric problems and has appealing statistical and computational properties. Moreover, it can effectively leverage additional structure such as feature sparsity.