  A multivariate quantile regression model with a factor structure is proposed
to study data with many responses of interest. The factor structure is allowed
to vary with the quantile levels, which makes our framework more flexible than
the classical factor models. The model is estimated with the nuclear norm
regularization in order to accommodate the high dimensionality of data, but the
incurred optimization problem can only be efficiently solved in an approximate
manner by off-the-shelf optimization methods. Such a scenario is often seen
when the empirical risk is non-smooth or the numerical procedure involves
expensive subroutines such as singular value decomposition. To ensure that the
approximate estimator accurately estimates the model, non-asymptotic bounds on
error of the the approximate estimator is established. For implementation, a
numerical procedure that provably marginalizes the approximate error is
proposed. The merits of our model and the proposed numerical procedures are
demonstrated through Monte Carlo experiments and an application to finance
involving a large pool of asset returns.
