Robust Signal Recovery for High-Dimensional Linear Log-Contrast Models with Compositional Covariates

Abstract

In this article, we propose a robust signal recovery method for high-dimensional linear log-contrast models, when the error distribution could be heavy-tailed and asymmetric. The proposed method is built on the Huber loss with ${\ell }_1$ penalization. We establish the ${\ell }_1$ and ${\ell }_2$ consistency for the resulting estimator. Under conditions analogous to the irrepresentability condition and the minimum signal strength condition, we prove that the signed support of the slope parameter vector can be recovered with high probability. The finite-sample behavior of the proposed method is evaluated through simulation studies, and applications to a GDP satisfaction dataset an HIV microbiome dataset are provided.

KEYWORDS:

• Compositional data
• Consistent estimation
• Huber loss
• Lasso
• Support recovery

Supplementary Materials

All proofs are provided in the supplementary materials.

Acknowledgments

The authors are grateful to the Editor, the Associate Editor and the anonymous reviewers for their professional review and constructive comments that lead to significant improvements in the article.

Funding

Additional information

Funding

The work of D. Han is supported by the National Natural Science Foundation of China (grant no. 12101330) and the Fundamental Research Funds for the Central Universities, Nankai University, 9920200110. The work of J. Huang is supported in part by the U.S. National Science Foundation grant DMS-1916199. The work of Y. Lin is partially supported by the Hong Kong Research Grants Council (grant nos. 14306219 and 14306620), the National Natural Science Foundation of China (grant no. 11961028) and Direct Grants for Research, The Chinese University of Hong Kong. The work of L. Liu is supported by NIH ULI TR002345. The work of L. Qu is supported by the National Natural Science Foundation of China (grant no. 12001219). The work of L. Sun is supported by the National Natural Science Foundation of China (grant no. 12171463).