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