Juan Liu

Wenwen Min, Juan Liu, Shihua Zhang

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 (GL1-SVD) and group L0-norm penalty (GL0-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 (OGL1-SVD) and overlapping group L0-norm penalty (OGL0-SVD). We first adopt an alternating iterative strategy to solve GL1-SVD based on a block coordinate descent method, and GL0-SVD based on a projection method. The key of solving OGL1-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 OGL0-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.

Wenwen Min, Juan Liu, Shihua Zhang

Given two data matrices $X$ and $Y$, sparse canonical correlation analysis (SCCA) is to seek two sparse canonical vectors $u$ and $v$ to maximize the correlation between $Xu$ and $Yv$. However, classical and sparse CCA models consider the contribution of all the samples of data matrices and thus cannot identify an underlying specific subset of samples. To this end, we propose a novel sparse weighted canonical correlation analysis (SWCCA), where weights are used for regularizing different samples. We solve the $L_0$-regularized SWCCA ($L_0$-SWCCA) using an alternating iterative algorithm. We apply $L_0$-SWCCA to synthetic data and real-world data to demonstrate its effectiveness and superiority compared to related methods. Lastly, we consider also SWCCA with different penalties like LASSO (Least absolute shrinkage and selection operator) and Group LASSO, and extend it for integrating more than three data matrices.

Wenwen Min, Juan Liu, Shihua Zhang

Molecular profiling data (e.g., gene expression) has been used for clinical risk prediction and biomarker discovery. However, it is necessary to integrate other prior knowledge like biological pathways or gene interaction networks to improve the predictive ability and biological interpretability of biomarkers. Here, we first introduce a general regularized Logistic Regression (LR) framework with regularized term $λ\|\bm{w}\|_1 + η\bm{w}^T\bm{M}\bm{w}$, which can reduce to different penalties, including Lasso, elastic net, and network-regularized terms with different $\bm{M}$. This framework can be easily solved in a unified manner by a cyclic coordinate descent algorithm which can avoid inverse matrix operation and accelerate the computing speed. However, if those estimated $\bm{w}_i$ and $\bm{w}_j$ have opposite signs, then the traditional network-regularized penalty may not perform well. To address it, we introduce a novel network-regularized sparse LR model with a new penalty $λ\|\bm{w}\|_1 + η|\bm{w}|^T\bm{M}|\bm{w}|$ to consider the difference between the absolute values of the coefficients. And we develop two efficient algorithms to solve it. Finally, we test our methods and compare them with the related ones using simulated and real data to show their efficiency.

Wenwen Min, Juan Liu, Shihua Zhang

Learning the "blocking" structure is a central challenge for high dimensional data (e.g., gene expression data). Recently, a sparse singular value decomposition (SVD) has been used as a biclustering tool to achieve this goal. However, this model ignores the structural information between variables (e.g., gene interaction graph). Although typical graph-regularized norm can incorporate such prior graph information to get accurate discovery and better interpretability, it fails to consider the opposite effect of variables with different signs. Motivated by the development of sparse coding and graph-regularized norm, we propose a novel sparse graph-regularized SVD as a powerful biclustering tool for analyzing high-dimensional data. The key of this method is to impose two penalties including a novel graph-regularized norm ($|\pmb{u}|\pmb{L}|\pmb{u}|$) and $L_0$-norm ($\|\pmb{u}\|_0$) on singular vectors to induce structural sparsity and enhance interpretability. We design an efficient Alternating Iterative Sparse Projection (AISP) algorithm to solve it. Finally, we apply our method and related ones to simulated and real data to show its efficiency in capturing natural blocking structures.