Approximate nonparametric quantile regression in reproducing kernel Hilbert spaces via random projection

Nonparametric quantile regression is a commonly used nonlinear quantile model. One general and popular approach is based on the use of kernels within a reproducing kernel Hilbert space (RKHS) framework, with the smoothing splines estimation as a special case. However, when the sample size n is large, the computational burden is heavy. Motivated by the recent advances in random projection for kernel nonparametric (mean) ridge regression (KRR), we consider an m-dimensional random projection approach for kernel quantile regression (KQR) with m≪n. We establish a theoretical result showing that the sketched KQR still achieves the minimax convergence rate when m is at least as large as the effective statistical dimension of the problem. Some Monte Carlo studies are carried out for illustration purposes.