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：本文提出了一种因子增强型稀疏吞吐(FAST)模型，该模型利用潜在因子和稀疏异质成分进行非参数回归。FAST模型一端连接因子模型，另一端连接稀疏非参数模型，涵盖了因子增强可加模型和稀疏低维非参数交互模型等结构化非参数模型，并适用于协变量不具备因子结构的情形。通过多样化投影作为潜在因子空间的估计，我们采用截断深度ReLU网络进行无正则化的非参数因子回归，以及使用非凸正则化的更一般FAST模型，分别得到因子增强型神经网络回归(FAR-NN)和FAST-NN估计量。我们证明，FAR-NN和FAST-NN估计量通过层次组合模型以非渐近极小化最优速率适应未知的低维结构。我们还研究了使用更具体神经网络架构的因子增强型稀疏可加模型的统计学习。我们的结果适用于无因子结构的弱依赖情形。在证明FAST-NN的主要技术结果时，我们建立了一个新的深度ReLU网络逼近结果，为神经网络理论基础做出了贡献。我们的理论和方法通过仿真研究以及宏观经济数据的应用得到了进一步验证。