  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.
