On the null space property of lq -minimization for 0

Abstract

The paper discusses the relationship between the null space property (NSP) and the lq-minimization in compressed sensing. Several versions of the null space property, that is, the lq stable NSP, the lq robust NSP, and the lq,p robust NSP for 0 <1 based on the standard lq NSP, are proposed, and their equivalent forms are derived. Consequently, reconstruction results for the lq-minimization can be derived easily under the NSP condition and its equivalent form. Finally, the lq NSP is extended to the lq-synthesis modeling and the mixed l2/lq-minimization, which deals with the dictionary-based sparse signals and the block sparse signals, respectively.