Robust Bilinear Probabilistic PCA Using a Matrix Variate t Distribution

The bilinear probabilistic principal component analysis (BPPCA) was introduced recently as a model-based dimension reduction technique on matrix data. However, BPPCA is based on the Gaussian assumption and hence is vulnerable to potential outlying matrix-valued observations. In this article, we present a new robust extension of BPPCA, called BPPCA using a matrix variate
${t}$
distribution (
${t}$
BPPCA), that is built upon a matrix variate
$t$
distribution. Like the multivariate
$t$
, this distribution offers an additional robustness tuning parameter, which can downweight outliers. By introducing a Gamma distributed latent weight variable, this distribution can be represented hierarchically. With this representation, two efficient accelerated expectation–maximization (EM)-like algorithms for parameter estimation are developed. Experiments on a number of synthetic and real datasets are conducted to understand
${t}$
BPPCA and compare with several closely related competitors, including its vector-based counterpart. The results reveal that
${t}$
BPPCA is generally more robust and accurate in the presence of outliers. Moreover, the expected latent weights under
${t}$
BPPCA can be effectively used for outliers’ detection, which is much more reliable than its vector-based counterpart due to its better robustness.