  Factor models are a class of powerful statistical models that have been
widely used to deal with dependent measurements that arise frequently from
various applications from genomics and neuroscience to economics and finance.
As data are collected at an ever-growing scale, statistical machine learning
faces some new challenges: high dimensionality, strong dependence among
observed variables, heavy-tailed variables and heterogeneity. High-dimensional
robust factor analysis serves as a powerful toolkit to conquer these
challenges.
  This paper gives a selective overview on recent advance on high-dimensional
factor models and their applications to statistics including Factor-Adjusted
Robust Model selection (FarmSelect) and Factor-Adjusted Robust Multiple testing
(FarmTest). We show that classical methods, especially principal component
analysis (PCA), can be tailored to many new problems and provide powerful tools
for statistical estimation and inference. We highlight PCA and its connections
to matrix perturbation theory, robust statistics, random projection, false
discovery rate, etc., and illustrate through several applications how insights
from these fields yield solutions to modern challenges. We also present
far-reaching connections between factor models and popular statistical learning
problems, including network analysis and low-rank matrix recovery.
