This paper focuses on minimizing a smooth function combined with a nonsmooth regularization term on a compact Riemannian submanifold embedded in the Euclidean space under a decentralized setting. Typically, there are two types of approaches at present for tackling such composite optimization problems. The first, subgradient-based approaches, rely on subgradient information of the objective function to update variables, achieving an iteration complexity of $O(\epsilon^{-4}\log^2(\epsilon^{-2}))$. The second, smoothing approaches, involve constructing a smooth approximation of the nonsmooth regularization term, resulting in an iteration complexity of $O(\epsilon^{-4})$. This paper proposes a proximal gradient type algorithm that fully exploits the composite structure. The global convergence to a stationary point is established with a significantly improved iteration complexity of $O(\epsilon^{-2})$. To validate the effectiveness and efficiency of our proposed method, we present numerical results from real-world applications, showcasing its superior performance compared to existing approaches.

Identification of interactions between molecular features (e.g. mutation, gene expression change) and gross phenotypes in diseases and other biological processes is one of the important challenges in genomic research. Popular approaches such as GSEA are limited to hypothesis tests of bivariate association. However, a specific phenotype is often dependent upon multiple molecular features. It is thus worth considering all possible interactions jointly for a more precise and realistic representation of the cellular network. In this article, a semiparametric copula model is developed to jointly model genotypes, pathways and phenotypes to accomplish this object. A two-step procedure for reconstruction of the network is described. Simulation studies indicate that the method is effective and accurate for the network reconstruction. Application using NCI60 cancer cell line data identifies several subsets of molecular features that jointly perform as the predictors of clinical phenotypes. The copula model is expected to have a broad impact on biomedical research, ranging from cancer treatment to disease prevention.