  Atomic norm minimization is a convex optimization framework to recover point
sources from a subset of their low-pass observations, or equivalently the
underlying frequencies of a spectrally-sparse signal. When the amplitudes of
the sources are positive, a positive atomic norm can be formulated, and exact
recovery can be ensured without imposing a separation between the sources, as
long as the number of observations is greater than the number of sources.
However, the classic formulation of the atomic norm requires to solve a
semidefinite program involving a linear matrix inequality of a size on the
order of the signal dimension, which can be prohibitive. In this letter, we
introduce a novel "compressed" semidefinite program, which involves a linear
matrix inequality of a reduced dimension on the order of the number of sources.
We guarantee the tightness of this program under certain conditions on the
operator involved in the dimensionality reduction. Finally, we apply the
proposed method to direction finding over sparse arrays based on second-order
statistics and achieve significant computational savings.
