Fast distribution to real regression

We study the problem of distribution to real regression, where one aims to regress a map-ping f that takes in a distribution input co-variate P ∈ I (for a non-parametric family of distributions I) and outputs a real-valued response Y = f(P) + . 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 Ω(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 in-stances N when evaluating new predictions after training; secondly, the Double-Basis es-timator is shown to have a fast rate of conver-gence for a general class of mappings f ∈ F.