  Principal component analysis (PCA) is one of the most widely used dimension
reduction and multivariate statistical techniques. From a probabilistic
perspective, PCA seeks a low-dimensional representation of data in the presence
of independent identical Gaussian noise. Probabilistic PCA (PPCA) and its
variants have been extensively studied for decades. Most of them assume the
underlying noise follows a certain independent identical distribution. However,
the noise in the real world is usually complicated and structured. To address
this challenge, some variants of PCA for data with non-IID noise have been
proposed. However, most of the existing methods only assume that the noise is
correlated in the feature space while there may exist two-way structured noise.
To this end, we propose a powerful and intuitive PCA method (MN-PCA) through
modeling the graphical noise by the matrix normal distribution, which enables
us to explore the structure of noise in both the feature space and the sample
space. MN-PCA obtains a low-rank representation of data and the structure of
noise simultaneously. And it can be explained as approximating data over the
generalized Mahalanobis distance. We develop two algorithms to solve this
model: one maximizes the regularized likelihood, the other exploits the
Wasserstein distance, which is more robust. Extensive experiments on various
data demonstrate their effectiveness.
