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.
A previous nonparametric FFR estimator was proposed in @cite_9 . @cite_9 attempts to perform FFR on a functional RKHS. That is, if we consider @math as a functional Hilbert space, where @math is such that @math , then @math is estimated by @math However, when each function is observed though @math noisy function evaluations this estimator will require the inversion of a @math matrix, which will be computationally infeasible for data-sets of even a modest size.
@cite_9
2127128771
This paper deals with functional regression, in which the input attributes as well as the response are functions. To deal with this problem, we develop a functional reproducing kernel Hilbert space approach; here, a kernel is an operator acting on a function and yielding a function. We demonstrate basic properties of these functional RKHS, as well as a representer theorem for this setting; we investigate the construction of kernels; we provide some experimental insight.