A New Algorithm Based on Linearized Bregman Iteration with Generalized Inverse for Compressed Sensing

Abstract

This paper aims to solve the basis pursuit problem $\min_{u\in\mathbb{R}^{n}}\{\|u\|_{1}: Au=f\}$ in compressed sensing when A is any matrix, there are two contributions in this paper. First, we provide a simplified iterative formula that combines original linearized Bregman iteration together with a soft threshold and iterative formula for generalized inverse of matrix $A\in\mathbb{R}^{m\times n}$. Furthermore, we also discuss its convergence. Compared to the original linearized Bregman method, the proposed simplified iterative scheme possesses obvious advantage. The workload and computing time are greatly reduced, when m≪n and $\operatorname{rank}(A)\ll m$, especially. But the requirement of high accuracy cannot be achieved. So, we need to select the proper number of inner loop to achieve the goal of balancing the workload and the accuracy of the simplified iterative formula. Second, we propose a new chaotic iterative algorithm based on the simplified iteration. Under the same iterations, the computing time of the simplified iteration with q=1 (q is the number of inner loops) is almost the same as that of the chaotic method, but the precision of the latter is better than that of the former because it utilized more information; and the accuracy of the chaotic method achieves that of A ^† linearized Bregman iteration, while the computing time of the former is less than one half of the latter. In conclusion, the calculating efficiency of our two methods as regards A ^† iteration is improved, and specially the chaotic iteration is more competitive. Numerical results on compressed sensing are presented that demonstrate that our methods can be significantly more effective than the original linearized Bregman iterations, even when matrix A is ill-conditioned.