Adaptive Sparse Estimation With Side Information

ABSTRACT

The article considers the problem of estimating a high-dimensional sparse parameter in the presence of side information that encodes the sparsity structure. We develop a general framework that involves first using an auxiliary sequence to capture the side information, and then incorporating the auxiliary sequence in inference to reduce the estimation risk. The proposed method, which carries out adaptive Stein’s unbiased risk estimate-thresholding using side information (ASUS), is shown to have robust performance and enjoy optimality properties. We develop new theories to characterize regimes in which ASUS far outperforms competitive shrinkage estimators, and establish precise conditions under which ASUS is asymptotically optimal. Simulation studies are conducted to show that ASUS substantially improves the performance of existing methods in many settings. The methodology is applied for analysis of data from single cell virology studies and microarray time course experiments. Supplementary materials for this article are available online.

KEYWORDS:

• Adaptive shrinkage estimation
• Higher order minimax risk
• Inference with side information
• Sparsity
• SURE shrinkage
• Two-sample inference

Supplementary Materials

This supplement contains a detailed description of the Auxiliary Screening procedure (Aux-Scr), proofs of the results in Section 2 and 3 of the main paper, additional simulation experiments, a real data analysis and an example that demonstrates a data driven procedure for choosing K.

Acknowledgments

We thank Ann Arvin and Nandini Sen for helpful discussions on the virology application. We thank the AE and two referees for the constructive suggestions that have greatly helped to improve the presentation of the article. In particular, we are grateful to an excellent comment from a referee that leads to the Bayesian interpretation of ASUS in Section 2.4.

Additional information

Funding

The research of WS was supported in part by NSF grants DMS-CAREER 1255406 and DMS-1712983. TB and GM were partially supported by NSF DMS-1811866 and by the Zumberge individual award from the University of Southern California‘s James H. Zumberge Faculty Research and Innovation Fund.