Fast Function to Function Regression

TL;DR

This paper introduces the Triple-Basis Estimator (3BE), a scalable nonparametric method for function-to-function regression that efficiently handles large datasets, with proven risk bounds and superior speed and accuracy.

Contribution

The paper presents the first scalable nonparametric estimator for function-to-function regression, enabling analysis of large datasets with improved speed and reduced error.

Findings

3BE achieves significant speed improvements over previous methods.

3BE provides competitive or better prediction accuracy.

Theoretical risk bounds for 3BE are derived.

Abstract

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.