Risk bounds for quantile trend filtering

Summary We study quantile trend filtering, a recently proposed method for nonparametric regression, with the goal of generalizing existing risk bounds usual trend-filtering estimators that perform mean regression. both penalized and constrained versions, order $r \geqslant 1$, univariate filtering. Our results show versions 1$ attain minimax rate up to logarithmic factors, when $(r-1)$th discrete derivative true vector quantiles belongs class bounded-variation signals. Moreover, we if is spline few polynomial pieces, then near-parametric convergence. Corresponding are known hold only errors sub-Gaussian. In contrast, our shown under minimal assumptions on error variables. particular, no moment needed heavy-tailed errors. proof techniques general, thus can potentially be used other regression methods. To illustrate this generality, employ obtain new multivariate total-variation denoising high-dimensional linear