Sparse orthogonal factor regression for high-dimensional multi-response generalized linear models

In the expansive domain of big data analytics, datasets usually contain a variety of data types, including numerical values, binary or categorical variables, and frequency counts. The inherent intricacies and high-dimensionality of such datasets pose a significant challenge in developing models that are not only accurate and valid but also have a high degree of interpretability. Building upon the latest innovations in multivariate regression techniques and sparse singular value decomposition (SSVD), we introduce the generalized sparse orthogonal factor regression (GeSOFAR) method. This novel methodology seamlessly integrates the sparse orthogonal factor analysis regression (SOFAR) into the broader spectrum of generalized linear models, thereby enhancing its applicability. To addresses the computational challenges associated with the log-likelihood functions in multivariate generalized linear models, our approach adopts both the majorization-minimization (MM) and block coordinate descent (BCD) algorithms. Furthermore, we propose an algorithmic strategy that leverages the augmented Lagrangian multiplier (ALM) method, which was originally formulated within the SOFAR framework, to streamline the computation process. From a theoretical perspective, we have established nonasymptotic error bounds for our procedure, delineating its theoretical merits such as sparsity, orthogonality, consistency, and low-rank structure. The efficacy of our GeSOFAR method is substantiated through an array of simulations and its successful application to a practical scenario, highlighting its robustness and utility in real-world data analysis contexts.