  The root-cause diagnostics of product quality defects in multistage
manufacturing processes often requires a joint identification of crucial stages
and process variables. To meet this requirement, this paper proposes a novel
penalized matrix regression methodology for two-dimensional variable selection.
The method regresses a scalar response variable against a matrix-based
predictor using a generalized linear model. The unknown regression coefficient
matrix is decomposed as a product of two factor matrices. The rows of the first
factor matrix and the columns of the second factor matrix are simultaneously
penalized to inspire sparsity. To estimate the parameters, we develop a block
coordinate proximal descent (BCPD) optimization algorithm, which cyclically
solves two convex sub-optimization problems. We have proved that the BCPD
algorithm always converges to a critical point with any initialization. In
addition, we have also proved that each of the sub-optimization problems has a
closed-form solution if the response variable follows a distribution whose
(negative) log-likelihood function has a Lipschitz continuous gradient. A
simulation study and a dataset from a real-world application are used to
validate the effectiveness of the proposed method.
