Sketched quantile additive functional regression

The quantile additive functional regression (QAFR) relates the response to the integral of F(t,X(t)) over t, where F is an unknown function and X(t) is the predictor in the form of a curve (function). This model incorporates functional linear quantile regression as a special case and the appearance of the integration as a smoothing operator makes regularization necessary to obtain a meaningful estimate as in typical inverse problems. It is conventional to perform ridge regression over the reproducing kernel Hilbert space. However, the computational complexity grows very fast when the problem scale gets large. We are thus motivated to consider the random sketching method as an approximation for this class of models. Sketching has become very popular recently and was previously used in some learning problems including kernel ridge regression. We established the optimal theoretical risk bound showing that the sketched estimator can converge as fast as the standard estimator. Numerical studies are carried out to examine the performance of sketched estimation.