  Matrix factor model has been growing popular in scientific fields such as
econometrics, which serves as a two-way dimension reduction tool for matrix
sequences. In this article, we for the first time propose the matrix elliptical
factor model, which can better depict the possible heavy-tailed property of
matrix-valued data especially in finance. Manifold Principle Component Analysis
(MPCA) is for the first time introduced to estimate the row/column loading
spaces. MPCA first performs Singular Value Decomposition (SVD)for each "local"
matrix observation and then averages the local estimated spaces across all
observations, while the existing ones such as 2-dimensional PCA first
integrates data across observations and then does eigenvalue decomposition of
the sample covariance matrices. We propose two versions of MPCA algorithms to
estimate the factor loading matrices robustly, without any moment constraints
on the factors and the idiosyncratic errors. Theoretical convergence rates of
the corresponding estimators of the factor loading matrices, factor score
matrices and common components matrices are derived under mild conditions. We
also propose robust estimators of the row/column factor numbers based on the
eigenvalue-ratio idea, which are proven to be consistent. Numerical studies and
real example on financial returns data check the flexibility of our model and
the validity of our MPCA methods.
