Sparse representations are at the core of many low-level signal processing procedures and are used by most pattern recognition algorithms to reduce the dimension of the search space. Structuring sparse representations fro pattern recognition applications requires taking into account invariants relatively to physical deformations such as rotation scaling or illumination. Sparsity, invariants and stability are conflicting requirements which is a source of open problems. Structured sparse representations with locally linear vector spaces are introduced for super-resolution inverse problems and pattern recognition.