Fast Distribution To Real Regression

We study the problem of distribution to real regression, where one aims to regress a mapping f that takes in a distribution input covariate P is an element of I (for a non-parametric family of distributions I) and outputs a real-valued response Y f(P) + epsilon. This setting was recently studied in [15], where the "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 Omega(N). 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 N 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 f is an element of F.