It is well known that microbiome sequencing data are naturally compositional and often have a complex subcomposition structure, as well as a lot of noise and some outliers. These characteristics make classical log-contrast regression difficult to apply, because it is sensitive to extreme values and cannot fully meet the linear constraints of compositional predictors. To tackle these issues, in this paper, we propose a robust log-contrast regression framework that incorporates the Huber loss function along with a Lasso penalty, which not only boosts resistance to outliers but also facilitates sparse variable selection. It is important to note that the general linear constraints, which are explicitly imposed, serve to preserve the algebraic coherence of compositional and subcompositional covariates. Under some mild assumptions, we theoretically prove that the proposed method achieves non-asymptotic guarantees in finite samples, comparable to those of Huber loss estimators, ensuring robustness. To implement the method efficiently, we develop a symmetric Gauss--Seidel-based alternating direction method of multipliers based on its dual formulation with a convergence guarantee, and it is also easy to implement in the sense that each subproblem admits a closed-form solution. To evaluate the statistical performance of the proposed method, we conduct a series of numerical experiments using simulated data and gut microbiome data, which show that the proposed method achieves better estimation accuracy, competitive prediction performance, and more reliable outlier identification than the benchmark method RobRegCC.

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