  This paper is concerned with estimating unknown multi-dimensional frequencies
from linear compressive measurements. This is accomplished by employing the
recently proposed atomic norm minimization framework to recover these
frequencies under a sparsity prior without imposing any grid restriction on
these frequencies. To this end, we give a rigorous derivation of an iterative
scheme called alternating direction of multipliers method, which is able to
incorporate multiple compressive snapshots from a multi-dimensional
superposition of complex harmonics. The key result here is how to formulate the
objective function minimized by this scheme and its partial derivatives, which
become hard to manage if the dimensionality of the frequencies is larger than
1. Moreover we demonstrate the performance of this approach in case of 3D line
spectral estimation and 2D DOA estimation with a synthetic antenna array.
