Abstract
With the application of regularization, sparse component analysis (SCA) becomes an effective approximate method for finding the sparsest solution of signal decomposition in an overcomplete dictionary. In this paper, the authors present some conditions on a family of regularization functions indexed by a hyperparameter so that these functions are applicable and can be effectively optimized in SCA. For a given signal, with these conditions it is proven that there exists at least one hyperparameter such that the global minimum of the corresponding regularization function can theoretically lead to the sparsest representation of the signal. Based on these propositions, a general principle is presented for the construction of regularization functions, and several kinds of function families are recommended for the purpose of sparse signal representation. The paper gives a numerical example that indicates that, for a synthesized signal, minimizing the regularization function proposed in this paper provides the correct sparse solution, whereas the method of basis pursuit fails.
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