Fast Function to Function Regression.

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 timeseries 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 datasets 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 data-sets with many instances. To the best of our knowledge, the 3BE is the first nonparametric FFR estimator that can scale to massive data-sets. We analyze the 3BE's risk and derive an upper-bound 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.