Group-Sparse SVD Models via <inline-formula><tex-math notation="LaTeX">$L_1$</tex-math><alternatives><mml:math><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="zhang-ieq1-2932063.gif"/></alternatives></inline-formula>- and <inline-formula><tex-math nota

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 (<inline-formula><tex-math notation="LaTeX">$GL_1$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>G</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="zhang-ieq3-2932063.gif"/></alternatives></inline-formula>-SVD) and group <inline-formula><tex-math notation="LaTeX">$L_0$</tex-math><alternatives><mml:math><mml:msub><mml:mi>L</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="zhang-ieq4-2932063.gif"/></alternatives></inline-formula>-norm penalty (<inline-formula><tex-math notation="LaTeX">$GL_0$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>G</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="zhang-ieq5-2932063.gif"/></alternatives></inline-formula>-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 (<inline-formula><tex-math notation="LaTeX">$OGL_1$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mi>G</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="zhang-ieq6-2932063.gif"/></alternatives></inline-formula>-SVD) and overlapping group <inline-formula><tex-math notation="LaTeX">$L_0$</tex-math><alternatives><mml:math><mml:msub><mml:mi>L</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="zhang-ieq7-2932063.gif"/></alternatives></inline-formula>-norm penalty (<inline-formula><tex-math notation="LaTeX">$OGL_0$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mi>G</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="zhang-ieq8-2932063.gif"/></alternatives></inline-formula>-SVD). We first adopt an alternating iterative strategy to solve <inline-formula><tex-math notation="LaTeX">$GL_1$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>G</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="zhang-ieq9-2932063.gif"/></alternatives></inline-formula>-SVD based on a block coordinate descent method, and <inline-formula><tex-math notation="LaTeX">$GL_0$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>G</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="zhang-ieq10-2932063.gif"/></alternatives></inline-formula>-SVD based on a projection method. The key of solving <inline-formula><tex-math notation="LaTeX">$OGL_1$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mi>G</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="zhang-ieq11-2932063.gif"/></alternatives></inline-formula>-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 <inline-formula><tex-math notation="LaTeX">$OGL_0$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mi>G</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="zhang-ieq12-2932063.gif"/></alternatives></inline-formula>-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.