PG-ADMM Algorithm Based On 1-Bit Compressing Sensing

Compressed sensing is able to restore high-dimensional signals from a small number of linear measurements,and by developing the sparsity of signals,compressible or sparse signals are obtained through linear measurement systems to obtain discrete samples of signals,and then reconstruct the signals through special tracking methods.1-Bit compressed sensing treats the measured value as a symbolic constraint,retains only the symbolic information of random measurement and further constraint optimization to restore the signal on the unit sphere,and then uses classical compressed sensing for reconstruction.This method performs much better than the classical compressed sensing reconstruction method even when the signal is relatively sparse.In this paper,a non-convex and non-smooth optimization model based on 1-Bit compressed sensing is proposed,which contains the hinge loss function and the l0 norm of the reconstructed signal.In the process of solving this problem,the quadratic penalty function of the objective function is first constructed,the first-order optimality condition is established by introducing the concept of P-stationary point of the penalty function,and the relationship between the P-stationary point and the local minimum point of the penalty function is proved.Then,the Proximal Gradient(PG)algorithm is used to solve the minimization problem of the penalty function,and the local convergence of the algorithm is proved.Finally,the Proximal Gradient Alternating Direction Method of Multipliers(PG-ADMM)algorithm is proposed based on the above theory,and the algorithm is divided into two steps.The first step is to use the PG algorithm to obtain the local approximate solution on a certain subspace of the penalty function,and the second step is to optimize it with the classical ADMM algorithm on the basis of the first step.