coefficient vector of the $k$th sinusoid. Following from [@tang2012compressed; @yang2014continuous; @yang2014exact], $\norm{\m{Y}}_{\cA,0}$ can be characterized as the following rank minimization problem: Throughout this paper we use the following identity whenever $\m{R}\in \bC^{N\times N}$ is positive semidefinite: The first constraint in (\[formu:atom0norm\]) imposes that $\m{Y}$ lies in the range space of a (Hermitian) Toeplitz matrix where $u_j$ is the $j$th entry of $\m{u}\in\bC^N$. The frequencies composing $\m{Y}$ are encoded in $T\sbra{\m{u}}$. Once an optimizer of $\m{u}$, say $\m{u}^*$, is obtained the frequencies can be retrieved from $T\sbra{\m{u}^*}$ using the Vandermonde decomposition lemma (see, e.g., [@stoica2005spectral]), which states that any PSD Toeplitz matrix $T\sbra{\m{u}^*}$ can be decomposed as where the order $K^*=\rank\sbra{T\sbra{\m{u}^*}}$ and $p_k^*>0$ (note that this decomposition is unique if $K^*<N$ and a computational method can be found in [@yang2015gridless Appendix A]). Therefore, by (\[formu:atom0norm\]) the CCS problem is reformulated as a LRMR problem in which the matrix $T\sbra{\m{u}}$ is Toeplitz and PSD and its range space contains $\m{Y}$.

The atomic $\ell_0$ norm exploits sparsity to the greatest extent possible; however, it is nonconvex and NP-hard to compute according to the rank minimization formulation and it thus encourages computationally feasible alternatives. In this spirit, the atomic ($\ell_1$) norm, denoted by $\atomn{\m{Y}}$, 