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Analysis sparse representation has recently emerged as an alternative approach to the synthesis sparse model. Most existing algorithms typically employ the -norm, which is generally NP-hard. Other existing algorithms employ the -norm to relax the -norm, which sometimes cannot promote adequate sparsity. Most of these existing algorithms focus on general signals and are not suitable for nonnegative signals. However, many signals are necessarily nonnegative such as spectral data. In this paper, we present a novel and efficient analysis dictionary learning algorithm for nonnegative signals with the determinant-type sparsity measure which is convex and differentiable. The analysis sparse representation can be cast in three subproblems, sparse coding, dictionary update, and signal update, because the determinant-type sparsity measure would result in a complex nonconvex optimization problem, which cannot be easily solved by standard convex optimization methods. Therefore, in the proposed algorithms, we use a difference of convex (DC) programming scheme for solving the nonconvex problem. According to our theoretical analysis and simulation study, the main advantage of the proposed algorithm is its greater dictionary learning efficiency, particularly compared with state-of-the-art algorithms. In addition, our proposed algorithm performs well in image denoising.

Analysis Sparse Representation for Nonnegative Signals Based on Determinant Measure by DC Programming