  We analyze the problem of regression when both input covariates and output
responses are functions from a nonparametric function class. Function to
function regression (FFR) covers a large range of interesting applications
including time-series prediction problems, and also more general tasks like
studying a mapping between two separate types of distributions. However,
previous nonparametric estimators for FFR type problems scale badly
computationally with the number of input/output pairs in a data-set. Given the
complexity of a mapping between general functions it may be necessary to
consider large data-sets in order to achieve a low estimation risk. To address
this issue, we develop a novel scalable nonparametric estimator, the
Triple-Basis Estimator (3BE), which is capable of operating over datasets with
many instances. To the best of our knowledge, the 3BE is the first
nonparametric FFR estimator that can scale to massive datasets. We analyze the
3BE's risk and derive an upperbound rate. Furthermore, we show an improvement
of several orders of magnitude in terms of prediction speed and a reduction in
error over previous estimators in various real-world data-sets.
