(or the total variation norm) [@chandrasekaran2012convex]. In addition, they proved that the frequencies can be recovered with infinite precision in the absence of noise once they are mutually separated by at least $\frac{4}{N}$. This theory was then extended to the cases of compressive data and MMVs by Tang *et al.* [@tang2012compressed] and the authors [@yang2014continuous; @yang2014exact], showing that the signal and the frequencies can be exactly recovered with high probability via atomic norm minimization (ANM) provided $M\geq O\sbra{K\ln K\ln N}$ and the same frequency separation condition holds. Other related papers include [@bhaskar2013atomic; @candes2013super; @tang2015near; @yang2014discretization; @yang2015gridless; @azais2014spike; @condat2015cadzow; @chen2014robust; @tan2014direction; @mishra2014off; @lu2015distributed]. While the atomic norm techniques completely eliminate grid mismatches of earlier grid-based sparse methods, a major problem is that the frequencies have to be sufficiently separated for successful recovery, prohibiting *high resolution*.[^2]

In this paper, we propose a high resolution gridless sparse method for signal and frequency recovery in CCS. Our method is motivated by the formulations and properties of the atomic $\ell_0$ norm and the atomic norm in [@yang2014continuous; @yang2014exact]. In particular, the atomic $\ell_0$ norm directly exploits sparsity and has no resolution limit but is NP hard to compute. To the contrary, as a convex relaxation 