A Research On Self-adaptive Gradient Algorithms For Compressed Sensing Problem

Compressed Sensing (CS) theory is a new signal sampling theory proposed in recent years for for the sparsesignal or compressible signal. It is a bold innovation of signal sampling theory which broke the bottleneck oftraditional Nyquist sampling law and have a very broad application prospects. The research of reconstructionalgorithms is a very important direction in CS field.In this paper,we focus on the survey of reconstruction algorithms. Specific content and innovations are asfollows:(1) Sparse reconstruction has been widely involved in signal processing and compressed sensing. In order torealise the task, one need to solve a l1-norm regularized least squares problem. Due to the nonsmooth objectivefunction, the resulting problem is challenging. This thesis presents a modified Polak-Ribie`re-Polyak (PRP)conjugate gradient method for sparse reconstruction with applications to compressed sensing. The constructionof the method consists of two main steps. Firstly, we reformulate the l1regularized least squares problem intoan equivalent nonlinear system of monotone equations. Then, we apply a projected PRP conjugate gradientmethod for solving the resulting system of monotone equations. The algorithm is easily implemented, in whichonly matrix-vector inner product is required at each step. Due to the low memory requirement of conjugategradient approaches, the proposed method is effective. Under suitable conditions, its global convergence resultis established. Numerical experiments show that the proposed method is practical and promising for sparsereconstruction.(2) In this thesis, we study a nonmonotone adaptive Barzilai-Borwein gradient algorithm for l1-norm mini-mization problems arising from sparse solution recovery in compressed sensing. At each iteration, the generatedsearch direction enjoys descent property and can be easily derived by minimizing a local approximal quadraticmodel and simultaneously taking the favorable structure of the l1-norm. Moreover, a nonmonotone line searchscheme is adopted to find a suitable stepsize along this direction. Under some conditions, the proposed algorith-m is shown to be globally convergent. Numerical results illustrate that the proposed method is promising andcompetitive with the existing algorithms NBBL1and the two-step IST (TwIST).