Research On Signal Recovery Theory Based On L₁-αl₂ Model

Compression sensing is a hot topic in the field of applied mathematics in recent years.It is a new sampling theory,which mainly considers to recover high-dimensional sparse signals from a few linear measurements by using various prior information of the signal itself.Thel₁-αl₂（0<α≤1）-based minimization method is a new and effective method for signal recovery in recent years.In this paper,thel₁-αl₂ model is used to provide sufficient conditions to ensure the stable recovery of sparse signals and the robust recovery of compressible signals with known partial supports,and the error estimates of signal recovery under different noise scenarios is obtained.In Chapter 2,sufficient conditions to ensure stable and robust signal recovery are given in the framework of restricted isometry property.In the first part,the conditions for accurate recovery of general signals and signals with partial prior support information are obtained by using the null space property.The second part specifically focuses on signals with known partial support sets and obtains the conditions and error estimates for ensuring signal robust recovery under two kinds of noise（l₂bounded noise,Dantzig Selector noise）.In the third part,a new method of decomposing non-sparse vectors into sparse vectors is used to obtain the conditions and error estimates for ensure the robust recovery of signals with known partial support under three kinds of noise（l₂ bounded noise,Dantzig Selector noise,impulse noise）.It is the first time to systematically study the problem of signal recovery with known partial support information under three kinds of noise.In Chapter 3,it is the first time to usel₁-αl₂ minimization model to discuss signal recovery in the framework of coherence.In Chapter 4,sufficient conditions to ensure stable and robust signal recovery are given in the framework of q-ratio CMSV.In this part,thel₁-αl₂ minimization model is used to obtain thel_q-norm andl₁-norm performance bounds of reconstruction errors in three noise cases based on q-ratio CMSV.Compared with the restricted isometry property and null space property,the sufficient condition based on q-ratio CMSV is more concise,and if the number of measurements is large enough,it can ensure that the sufficient condition has a higher probability for sub-Gaussian matrix and a class of structured random matrix.