Sequential Scaled Sparse Factor Regression

Abstract

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 article, 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.

Keywords:

• Big data
• Latent factors
• Scalability
• Sparse reduced-rank regression
• Stock short interest analysis
• Tuning insensitiveness

Supplementary Materials

In the supplementary materials, we give the technical proofs for Proposition 1, Corollary 1 and Theorems 1–2.

Natural Science Foundation of Anhui Province;

Acknowledgments

The authors would like to thank Ruipeng Dong for helpful discussions.

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

¹ The Gaussian assumption is not essential and we will show the validity of the proposed method under sub-Gaussian errors in Section 3.

Additional information

Funding

This work was supported by National Natural Science Foundation of China-72071187, 11601501, 11671374, 71731010, and 71921001, Anhui Provincial Natural Science Foundation-1708085QA02, and Fundamental Research Funds for the Central Universities-WK2040000027, WK2040160028.