  Robust Principal Component Analysis (RPCA) is a widely used method for
recovering low-rank structure from data matrices corrupted by significant and
sparse outliers. These corruptions may arise from occlusions, malicious
tampering, or other causes for anomalies, and the joint identification of such
corruptions with low-rank background is critical for process monitoring and
diagnosis. However, existing RPCA methods and their extensions largely do not
account for the underlying probabilistic distribution for the data matrices,
which in many applications are known and can be highly non-Gaussian. We thus
propose a new method called Robust Principal Component Analysis for Exponential
Family distributions ($e^{\text{RPCA}}$), which can perform the desired
decomposition into low-rank and sparse matrices when such a distribution falls
within the exponential family. We present a novel alternating direction method
of multiplier optimization algorithm for efficient $e^{\text{RPCA}}$
decomposition. The effectiveness of $e^{\text{RPCA}}$ is then demonstrated in
two applications: the first for steel sheet defect detection, and the second
for crime activity monitoring in the Atlanta metropolitan area.
