Rank-$k$ 2-D Multinomial Logistic Regression for Matrix Data Classification

The amount of matrix data has increased rapidly nowadays. How to classify matrix data efficiently is an important issue. In this paper, by discovering the shortages of 2-D linear discriminant analysis and 2-D logistic regression, a novel 2-D framework named rank-<inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> 2-D multinomial logistic regression (2DMLR-RK) is proposed. The 2DMLR-RK is designed for a multiclass matrix classification problem. In the proposed framework, each category is modeled by a left projection matrix and a right projection matrix with rank <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>. The left projection matrices capture the row information of matrix data, and the right projection matrices acquire the column information. We identify the parameter <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> plays the role of balancing the capacity of learning and generalization of the 2DMLR-RK. In addition, we develop an effective framework for solving the proposed nonconvex optimization problem. The convergence, initialization, and computational complexity are discussed. Extensive experiments on various types of data sets are conducted. Comparing with 1-D methods, 2DMLR-RK not only achieves a better classification accuracy, but also costs less computation time. Comparing with other state-of-the-art 2-D methods, the 2DMLR-RK achieves a better performance for matrix data classification.