Compressed Data Separation via ℓ_(q)-Split Analysis with ℓ_(∞)-Constraint

In this paper,we study compressed data separation(CDS)problem,i.e.,sparse data separation from a few linear random measurements.We propose the nonconvex ℓ_(q)-split analysis with ℓ_(∞)-constraint and 0<q≤1.We call the algorithm ℓ_(q)-split-analysis Dantzig selector(ℓ_(q)-split-analysis DS).We show that the two distinct subcomponents that are approximately sparse in terms of two different dictionaries could be stably approximated via the ℓ_(q)-split-analysis DS,provided that the measurement matrix satisfies either a classical D-RIP(Restricted Isometry Property with respect to Dictionaries and ℓ_(2) norm)or a relatively new(D,q)-RIP(RIP with respect to Dictionaries and ℓ_(q)-quasi norm)condition and the two different dictionaries satisfy a mutual coherence condition between them.For the Gaussian random measurements,the measurement number needed for the(D,q)-RIP condition is far less than those needed for the D-RIP condition and the(D,1)-RIP condition when q is small enough.