k-sparse vector recovery via Truncated l₁ - l₂ local minimization

This article mainly solves the following model, min parallel to x(Gamma x,t)(c)parallel to(1) - parallel to x(Gamma x,t)(c)parallel to(2) subject to Ax = y, where Gamma(x,t) subset of [n] represents the index of the maximum number of t elements in x after taking the absolute value. We call this model Truncated l(1) - l(2) model. We mainly deal with the recovery of unknown signals under the condition of vertical bar supp(x)vertical bar > t, sigma(t)(x) > sigma(t+1)(x), where sigma(t)(x) represents the t largest number in |x|. Firstly, we give the necessary and sufficient condition for recovering the fixed unknown signal satisfying the above two conditions via Truncated l(1) - l(2) local minimization. Then, according to this condition, we give the necessary and sufficient conditions to recovering for all unknown signals satisfying the above two conditions via Truncated l(1) - l(2) local minimization. Compared with N. Bi's recent proposed condition in Bi and Tang (Appl Comput Harmon Anal 56:337-350, 2022), we will show that our condition is weaker and the detail of such discussion is in Remark 3 of the manuscript. Then, we give the algorithm of Truncated l(1) - l(2) model. According to this algorithm, we do data experiments and the data experiments show that the recovery rate of Truncated l(1) - l(2) is better than that of model l(1) - l(2).