Research On Fast Sparse Bayesian Learning Algorithms

The Shannon-Nyquist sampling law has always been an important law that traditional signal acquisition needs to follow.The compressive sensing theory has broken through this bottleneck by compressively sampling signals at sub-Nyquist sampling rates and then recovering the original sparse signals through sparse reconstruction algorithms.This has greatly reduced the hardware and software pressure in data processing.The Sparse Bayesian Learning(SBL)framework completes the selection of relevant vectors and the reconstruction of sparse signals through posterior Bayesian inference under the prior assumption with sparse-inducing distributions,and it has received widespread attention in the field of compressive sensing due to the advantages such as no need to know the sparsity and higher sparsity of the reconstruction results.However,the traditional SBL algorithms need to frequently perform matrix inversion in the inference of posterior distribution and iterative update of hyperparameters,making it unsuitable for real-time processing of highdimensional data.Therefore,it has become an urgent problem to improve the SBL algorithms so that they have wider application scenarios and faster processing speed in highdimensional data processing.Based on the background mentioned above,this thesis focuses on the research of fast SBL algorithms.The main research contents of this thesis are as follows:Firstly,this thesis has conducted research on the SBL algorithms based on the type-Ⅱmarginal likelihood maximization and the variational inference.Through the analysis of hierarchical Bayesian probability models,the reasons why the SBL algorithm can obtain sparse solutions are analyzed.The computational complexity of the SBL algorithm is also discussed,and the feasible technical routes for accelerating the reconstruction process are pointed out.Secondly,to deal with the complexity issue of multi-task sparse reconstruction in both simultaneous sparse approximation and distributed compressive sensing,this thesis extends the fast inverse-free SBL algorithm from the single-task method to a multi-task version.By fully utilizing the correlations among different tasks,and transforming the matrix to be inverted into a diagonal matrix using a relaxed evidence lower bound,the new algorithm accelerates the inversion operation of the matrix and reduces reconstruction complexity.Simulation analysis shows that the proposed method has significant advantages in both signal reconstruction speed and reconstruction quality.Finally,in order to solve the problem that fast marginal likelihood maximization based SBL algorithm is sensitive to the initialization,which leads to unstable reconstruction performance,this thesis has improved its signal model by adding a noise precision parameter is added to the hierarchical prior model of the source vector.The noise precision parameter is integrated out to avoid the influence of the stability of the algorithm.Analysis shows that the new model can induce a prior distribution promoting higher sparsity for the source vector so the reconstruction result based on this model is sparser.Simulation experiments demonstrate that the proposed algorithm has significant advantages in both running speed and sparsity in high-dimensional sparse signal reconstruction,and can also achieve excellent reconstruction performance in sparse recovery of images.