Recent work in signal processing and statistics have focused on defining new regularization functions, which not only induce sparsity of the solution, but also take into account the structure of the problem. 1-7 We present in this paper a class of convex penalties introduced in the machine learning community, which take the form of a sum of ℓ 2 -and ℓ ∞ -norms over groups of variables. They extend the classical group-sparsity regularization 8-10 in the sense that the groups possibly overlap, allowing more flexibility in the group design. We review efficient optimization methods to deal with the corresponding inverse problems, [11] [12] [13] and their application to the problem of learning dictionaries of natural image patches: 14-18 On the one hand, dictionary learning has indeed proven effective for various signal processing tasks. 17, 19 On the other hand, structured sparsity provides a natural framework for modeling dependencies between dictionary elements. We thus consider a structured sparse regularization to learn dictionaries embedded in a particular structure, for instance a tree 11 or a two-dimensional grid. 20 In the latter case, the results we obtain are similar to the dictionaries produced by topographic independent component analysis.
Mairal et al. REF also considered a structured sparsity-inducing regularization and used it to learn dictionaries embedded in a particular structure.
Learning Hierarchical and Topographic Dictionaries with Structured Sparsity
SPIE Wavelets and Sparsity XIV 81381P (2011)
Engineering Computer Science