  Composite quantile regression has been used to obtain robust estimators of
regression coefficients in linear models with good statistical efficiency. By
revealing an intrinsic link between the composite quantile regression loss
function and the Wasserstein distance from the residuals to the set of
quantiles, we establish a generalization of the composite quantile regression
to the multiple-output settings. Theoretical convergence rates of the proposed
estimator are derived both under the setting where the additive error possesses
only a finite $\ell$-th moment (for $\ell > 2$) and where it exhibits a
sub-Weibull tail. In doing so, we develop novel techniques for analyzing the
M-estimation problem that involves Wasserstein-distance in the loss. Numerical
studies confirm the practical effectiveness of our proposed procedure.
