  As is known, factor analysis is a popular method to reduce dimension for
high-dimensional data. For matrix data, the dimension reduction can be more
effectively achieved through both row and column directions. In this paper, we
introduce a threshold factor models to analyze matrix-valued high-dimensional
time series data. The factor loadings are allowed to switch between regimes,
controlling by a threshold variable. The estimation methods for loading spaces,
threshold value, and the number of factors are proposed. The asymptotic
properties of these estimators are investigated. Not only the strengths of
thresholding and factors, but also their interactions from different directions
and different regimes play an important role on the estimation performance.
When the thresholding and factors are all strong across regimes, the estimation
is immune to the impact that the increase of dimension brings, which breaks the
curse of dimensionality. When the thresholding in two directions and factors
across regimes have different levels of strength, we show that estimators for
loadings and threshold value experience 'helping' effects against the curse of
dimensionality. We also discover that even when the numbers of factors are
overestimated, the estimators are still consistent. The proposed methods are
illustrated with both simulated and real examples.
