Sequential scaled sparse factor regression

Large-scale association analysis between multivariate responses and predictors is of great practical importance, as exemplified by modern business applications including social media marketing and crisis management. Despite the rapid methodological advances, how to obtain scalable estimators with free tuning of the regularization parameters remains unclear under general noise covariance structures. In this paper, we develop a new methodology called sequential scaled sparse factor regression (SESS) based on a new viewpoint that the problem of recovering a jointly low-rank and sparse regression coefficient matrix can be decomposed into several univariate response sparse regressions through regular eigenvalue decomposition. It combines the strengths of sequential estimation and scaled sparse regression, thus sharing the scalability and the tuning free property for sparsity parameters inherited from the two approaches. The stepwise convex formulation, sequential factor regression framework, and tuning insensitiveness make SESS highly scalable for big data applications. Comprehensive theoretical justifications with new insights into high-dimensional multi-response regressions are also provided. We demonstrate the scalability and effectiveness of the proposed method by simulation studies and stock short interest data analysis.