Abstract-We develop an efficient learning framework to construct signal dictionaries for sparse representation by selecting the dictionary columns from multiple candidate bases. By sparse, we mean that only a few dictionary elements, compared to the ambient signal dimension, can exactly represent or well-approximate the signals of interest. We formulate both the selection of the dictionary columns and the sparse representation of signals as a joint combinatorial optimization problem. The proposed combinatorial objective maximizes variance reduction over the set of training signals by constraining the size of the dictionary as well as the number of dictionary columns that can be used to represent each signal. We show that if the available dictionary column vectors are incoherent, our objective function satisfies approximate submodularity. We exploit this property to develop SDS OMP and SDS MA , two greedy algorithms with approximation guarantees. We also describe how our learning framework enables dictionary selection for structured sparse representations, e.g., where the sparse coefficients occur in restricted patterns. We evaluate our approach on synthetic signals and natural images for representation and inpainting problems.
The greedy based methods are more straightforward, for example, Cevher and Krause REF formulate the dictionary columns selection and sparse representation of signals as a joint combinatorial problem and propose a dictionary selection model.
Greedy Dictionary Selection for Sparse Representation
IEEE Journal of Selected Topics in Signal Processing
IEEE Journal of Selected Topics in Signal Processing
Computer Science