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.

中文翻译：

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正源的压缩超分辨率

原子范数最小化是一个凸优化框架，用于从低通观测的子集中恢复点源，或者等效地从频谱稀疏信号的基础频率中恢复。当源的振幅为正时，可以制定正的原子范数，只要观测数大于源数，就可以在不强加源之间分离的情况下确保精确恢复。然而，原子范数的经典公式需要求解一个涉及信号维度数量级的线性矩阵不等式的半定程序，这可能是令人望而却步的。在这封信中，我们介绍了一种新颖的“压缩”半定程序，它涉及一个线性矩阵不等式，其维度为源数量级的缩减。我们保证这个程序在一定条件下对参与降维的算子的严密性。最后，我们将所提出的方法应用于基于二阶统计的稀疏数组的测向，并实现了显着的计算节省。