A multivariate quantile regression model with a factor structure is proposed to study data with multivariate responses with covariates. The factor structure is allowed to vary with the quantile levels, which is more flexible than the classical factor models. Assuming the number of factors is small, and the number of responses and the input variables are growing with the sample size, the model is estimated with the nuclear norm regularization. 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 loss is nonsmooth or the numerical procedure involves expensive subroutines, for example, singular value decomposition. To show that the approximate estimator is still statistically accurate, we establish a nonasymptotic bound on the Frobenius risk and prediction risk. For implementation, a numerical procedure that provably marginalizes the approximation error is proposed. The merits of our model and the proposed numerical procedures are demonstrated through the Monte Carlo simulation and an application to finance involving a large pool of asset returns.

中文翻译：

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可分解的多任务分位数回归

提出了一种具有因子结构的多变量分位数回归模型来研究具有协变量的多变量响应的数据。因子结构允许随分位数水平变化，比经典因子模型更灵活。假设因子的数量很少，并且响应的数量和输入变量的数量随着样本量的增加而增长，则使用核范数正则化来估计模型。产生的优化问题只能通过现成的优化方法以近似的方式有效地解决。当经验损失不平滑或数值过程涉及昂贵的子程序（例如奇异值分解）时，通常会看到这种情况。为了表明近似估计量在统计上仍然准确，我们建立了 Frobenius 风险和预测风险的非渐近界。为了实现，提出了一种可证明将近似误差边缘化的数值过程。我们的模型和提议的数值程序的优点通过蒙特卡罗模拟和涉及大量资产回报的金融应用得到证明。