  Traditional nonparametric estimation methods often lead to a slow convergence
rate in large dimensions and require unrealistically enormous sizes of datasets
for reliable conclusions. We develop an approach based on mixed gradients,
either observed or estimated, to effectively estimate the function at
near-parametric convergence rates. The novel approach and computational
algorithm could lead to methods useful to practitioners in many areas of
science and engineering. Our theoretical results reveal a behavior universal to
this class of nonparametric estimation problems. We explore a general setting
involving tensor product spaces and build upon the smoothing spline analysis of
variance (SS-ANOVA) framework. For $d$-dimensional models under full
interaction, the optimal rates with gradient information on $p$ covariates are
identical to those for the $(d-p)$-interaction models without gradients and,
therefore, the models are immune to the "curse of interaction". For additive
models, the optimal rates using gradient information are root-$n$, thus
achieving the "parametric rate". We demonstrate aspects of the theoretical
results through synthetic and real data applications.
