A null space analysis of the ℓ1ℓ1-synthesis method in dictionary-based compressed sensing

An interesting topic in compressed sensing aims to recover signals with sparse representations in a dictionary. Recently the performance of the ℓ1ℓ1-analysis method has been a focus, while some fundamental problems for the ℓ1ℓ1-synthesis method are still unsolved. For example, what are the conditions for it to stably recover compressible signals under noise? Do coherent dictionaries allow the existence of sensing matrices that guarantee good performances of the ℓ1ℓ1-synthesis method? To answer these questions, we build up a framework for the ℓ1ℓ1-synthesis method. In particular, we propose a dictionary-based null space property (D-NSP) which, to the best of our knowledge, is the first sufficient and necessary condition for the success of ℓ1ℓ1-synthesis without measurement noise. With this new property, we show that when the dictionary D is full spark, it cannot be too coherent otherwise the ℓ1ℓ1-synthesis method fails for all sensing matrices. We also prove that in the real case, D-NSP is equivalent to the stability of ℓ1ℓ1-synthesis under noise.