  Factor Analysis based on multivariate $t$ distribution ($t$fa) is a useful
robust tool for extracting common factors on heavy-tailed or contaminated data.
However, $t$fa is only applicable to vector data. When $t$fa is applied to
matrix data, it is common to first vectorize the matrix observations. This
introduces two challenges for $t$fa: (i) the inherent matrix structure of the
data is broken, and (ii) robustness may be lost, as vectorized matrix data
typically results in a high data dimension, which could easily lead to the
breakdown of $t$fa. To address these issues, starting from the intrinsic matrix
structure of matrix data, a novel robust factor analysis model, namely bilinear
factor analysis built on the matrix-variate $t$ distribution ($t$bfa), is
proposed in this paper. The novelty is that it is capable to simultaneously
extract common factors for both row and column variables of interest on
heavy-tailed or contaminated matrix data. Two efficient algorithms for maximum
likelihood estimation of $t$bfa are developed. Closed-form expression for the
Fisher information matrix to calculate the accuracy of parameter estimates are
derived. Empirical studies are conducted to understand the proposed $t$bfa
model and compare with related competitors. The results demonstrate the
superiority and practicality of $t$bfa. Importantly, $t$bfa exhibits a
significantly higher breakdown point than $t$fa, making it more suitable for
matrix data.
