A projected gradient method for αℓ1−βℓ2 sparsity regularization**The work of this author was supported by the Fundamental Research Funds for the Central Universities (no. 2572018BC02), Heilongjiang Postdoctoral Research Developmental Fund (no. LBH-Q16008), the National Nature Science Foundation of China (no. 41304093). (3rd December 2020) - Ding, Liang; Han, Weimin
Abstract: The non-convex α ‖ ⋅ ‖ ℓ 1 − β ‖ ⋅ ‖ ℓ 2 ( α ⩾ β ⩾ 0 ) regularization is a new approach for sparse recovery. A minimizer of the α ‖ ⋅ ‖ ℓ 1 − β ‖ ⋅ ‖ ℓ 2 regularized function can be computed by applying the ST-( αℓ 1 − βℓ 2 ) algorithm which is similar to the classical iterative soft thresholding algorithm (ISTA). It is known that ISTA converges quite slowly, and a faster alternative to ISTA is the projected gradient (PG) method. However, the conventional PG method is limited to solve problems with the classical ℓ 1 sparsity regularization. In this paper, we present two accelerated alternatives to the ST-( αℓ 1 − βℓ 2 ) algorithm by extending the PG method to the non-convex α ‖ ⋅ ‖ ℓ 1 − β ‖ ⋅ ‖ ℓ 2 sparsity regularization. Moreover, we discuss a strategy to determine the radius R of the ℓ 1 -ball constraint by Morozov's discrepancy principle. Numerical results are reported to illustrate the efficiency of the proposed approach.