  We address the problem of defining a group sparse formulation for Principal
Components Analysis (PCA) - or its equivalent formulations as Low Rank
approximation or Dictionary Learning problems - which achieves a compromise
between maximizing the variance explained by the components and promoting
sparsity of the loadings. So we propose first a new definition of the variance
explained by non necessarily orthogonal components, which is optimal in some
aspect and compatible with the principal components situation. Then we use a
specific regularization of this variance by the group-$\ell_{1}$ norm to define
a Group Sparse Maximum Variance (GSMV) formulation of PCA. The GSMV formulation
achieves our objective by construction, and has the nice property that the
inner non smooth optimization problem can be solved analytically, thus reducing
GSMV to the maximization of a smooth and convex function under unit norm and
orthogonality constraints, which generalizes Journee et al. (2010) to group
sparsity. Numerical comparison with deflation on synthetic data shows that GSMV
produces steadily slightly better and more robust results for the retrieval of
hidden sparse structures, and is about three times faster on these examples.
Application to real data shows the interest of group sparsity for variables
selection in PCA of mixed data (categorical/numerical) .
