Fast Function to Function Regression

We analyze the problem of regression when both input covariates and output responses are func-tions from a nonparametric function class. Func-tion to function regression (FFR) covers a large range of interesting applications including time-series prediction problems, and also more gen-eral tasks like studying a mapping between two separate types of distributions. However, previ-ous nonparametric estimators for FFR type prob-lems scale badly computationally with the num-ber of input/output pairs in a data-set. Given the complexity of a mapping between general func-tions 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 Esti-mator (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 upper-bound rate. Furthermore, we show an improve-ment of several orders of magnitude in terms of prediction speed and a reduction in error over previous estimators in various real-world data-sets.