This paper introduces a Factor Augmented Sparse Throughput (FAST) model that utilizes both latent factors and sparse idiosyncratic components for nonparametric regression. The FAST model bridges factor models on one end and sparse nonparametric models on the other end. It encompasses structured nonparametric models such as factor augmented additive models and sparse low-dimensional nonparametric interaction models and covers the cases where the covariates do not admit factor structures. Via diversified projections as estimation of latent factor space, we employ truncated deep ReLU networks to nonparametric factor regression without regularization and to a more general FAST model using nonconvex regularization, resulting in factor augmented regression using neural network (FAR-NN) and FAST-NN estimators respectively. We show that FAR-NN and FAST-NN estimators adapt to the unknown low-dimensional structure using hierarchical composition models in nonasymptotic minimax rates. We also study statistical learning for the factor augmented sparse additive model using a more specific neural network architecture. Our results are applicable to the weak dependent cases without factor structures. In proving the main technical result for FAST-NN, we establish a new deep ReLU network approximation result that contributes to the foundation of neural network theory. Our theory and methods are further supported by simulation studies and an application to macroeconomic data.

翻译：本文提出一种因子增强稀疏吞吐（FAST）模型，该模型同时利用潜在因子与稀疏异质成分进行非参数回归。FAST模型一端衔接因子模型，另一端衔接稀疏非参数模型，涵盖了因子增强加性模型与稀疏低维非参数交互模型等结构化非参数模型，并适用于协变量不服从因子结构的情形。通过多样化投影估计潜在因子空间，我们采用截断深度ReLU网络分别实现无正则化的非参数因子回归以及基于非凸正则化的更一般FAST模型，从而得到因子增强神经网络回归（FAR-NN）与FAST-NN两类估计量。研究表明，基于层级组合模型，FAR-NN与FAST-NN估计量在非渐近极小化最优速率下能够自适应于未知低维结构。同时，我们利用更具针对性的神经网络架构，研究了因子增强稀疏加性模型的统计学习问题。本方法适用于无因子结构的弱相依情形。在FAST-NN主要技术结果的证明中，我们建立了新的深度ReLU网络逼近结果，为神经网络理论奠定了基础。通过仿真实验及宏观经济数据的应用，进一步验证了所提出理论与方法的有效性。