Group-sparse SVD Models via $L_1$- and $L_0$-norm Penalties and Their Applications in Biological Data

Abstract

Sparse Singular Value Decomposition (SVD) models have been proposed for biclustering high dimensional gene expression data to identify block patterns with similar expressions. However, these models do not take into account prior group effects upon variable selection. To this end, we first propose group-sparse SVD models with group Lasso ( $GL_1$ G L 1 -SVD) and group $L_0$ L 0 -norm penalty ( $GL_0$ G L 0 -SVD) for non-overlapping group structure of variables. However, such group-sparse SVD models limit their applicability in some problems with overlapping structure. Thus, we also propose two group-sparse SVD models with overlapping group Lasso ( $OGL_1$ O G L 1 -SVD) and overlapping group $L_0$ L 0 -norm penalty ( $OGL_0$ O G L 0 -SVD). We first adopt an alternating iterative strategy to solve $GL_1$ G L 1 -SVD based on a block coordinate descent method, and $GL_0$ G L 0 -SVD based on a projection method. The key of solving $OGL_1$ O G L 1 -SVD is a proximal operator with overlapping group Lasso penalty. We employ an alternating direction method of multipliers (ADMM) to solve the proximal operator. Similarly, we develop an approximate method to solve $OGL_0$ O G L 0 -SVD. Applications of these methods and comparison with competing ones using simulated data demonstrate their effectiveness. Extensive applications of them onto several real gene expression data with gene prior group knowledge identify some biologically interpretable gene modules.