We study the problem of distribution to real-value regression, where one aims to regress a mapping @math that takes in a distribution input covariate @math (for a non-parametric family of distributions @math ) and outputs a real-valued response @math . This setting was recently studied, and a "Kernel-Kernel" estimator was introduced and shown to have a polynomial rate of convergence. However, evaluating a new prediction with the Kernel-Kernel estimator scales as @math . This causes the difficult situation where a large amount of data may be necessary for a low estimation risk, but the computation cost of estimation becomes infeasible when the data-set is too large. To this end, we propose the Double-Basis estimator, which looks to alleviate this big data problem in two ways: first, the Double-Basis estimator is shown to have a computation complexity that is independent of the number of of instances @math when evaluating new predictions after training; secondly, the Double-Basis estimator is shown to have a fast rate of convergence for a general class of mappings @math .
DRR is related to the functional analysis, where one regresses a mapping whose input domain are functions @cite_12 . However, the objects DRR works over--distributions and their pdfs--are inferred through sets of samples drawn from the objects, with finite sizes. In functional analysis, the functions are inferred through observations of @math pairs that are often taken to be an arbitrarily dense grid in the domain of the functions. For a comprehensive survey in functional analysis see @cite_12 @cite_11 . Also, recently @cite_15 studied the problem of distribution to distribution regression, where both input and output covariates are distributions.
@cite_15 @cite_12 @cite_11
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Introduction to functional nonparametric statistics.- Some functional datasets and associated statistical problematics.- What is a well adapted space for functional data?.- Local weighting of functional variables.- Functional nonparametric prediction methodologies.- Some selected asymptotics.- Computational issues.- Nonparametric supervised classification for functional data.- Nonparametric unsupervised classification for functional data.- Mixing, nonparametric and functional statistics.- Some selected asymptotics.- Application to continuous time processes prediction.- Small ball probabilities, semi-metric spaces and nonparametric statistics.- Conclusion and perspectives. Introduction.- Life Course Data in Criminology.- The Nondurable Goods Index.- Bone Shapes from a Paleopathology Study.- Modeling Reaction Time Distributions.- Zooming in on Human Growth.- Time Warping Handwriting and Weather Records.- How do Bone Shapes Indicate Arthritis?- Functional Models for Test Items.- Predicting Lip Acceleration from Electromyography.- Variable Seasonal Trend in the Goods Index.- The Dynamics of Handwriting Printed Characters.- A Differential Equation for Juggling.