Matrix regression heterogeneity analysis

The development of modern science and technology has facilitated the collection of a large amount of matrix data in fields such as biomedicine. Matrix data modeling has been extensively studied, which advances from the naive approach of flattening the matrix into a vector. However, existing matrix modeling methods mainly focus on homogeneous data, failing to handle the data heterogeneity frequently encountered in the biomedical field, where samples from the same study belong to several underlying subgroups, and different subgroups follow different models. In this paper, we focus on regression-based heterogeneity analysis. We propose a matrix data heterogeneity analysis framework, by combining matrix bilinear sparse decomposition and penalized fusion techniques, which enables data-driven subgroup detection, including determining the number of subgroups and subgrouping membership. A rigorous theoretical analysis is conducted, including asymptotic consistency in terms of subgroup detection, the number of subgroups, and regression coefficients. Numerous numerical studies based on simulated and real data have been constructed, showcasing the superior performance of the proposed method in analyzing matrix heterogeneous data.