Analysis of weighted l̳₁-minimization for model based compressed sensing

Abstract

The central problem of Compressed Sensing is to recover a sparse signal from fewer measurements than its ambient dimension. Recent results by Donoho, and Candes and Tao giving theoretical guarantees that ( 1-minimization succeeds in recovering the signal in a large number of cases have stirred up much interest in this topic. Subsequent results followed, where prior information was imposed on the sparse signal and algorithms were proposed and analyzed to incorporate this prior information. In[13] Xu suggested the use of weighted l₁-minimization in the case where the additional prior information is probabilistic in nature for a relatively simple probabilistic model. In this thesis, we exploit the techniques developed in [13] to extend the analysis to a more general class of probabilistic models, where the probabilities are evaluations of a continuous function at uniformly spaced points in a given interval. For this case, we use weights which have a similar characterization . We demonstrate our techniques through numerical computations for a certain class of weights and compare some of our results with empirical data obtained through simulations.