Soft Recovery With General Atomic Norms
Provides a general theoretical framework for signal recovery via atomic norm minimization, applicable to infinite-dimensional Hilbert spaces, which is of interest to researchers in signal processing and compressed sensing.
This paper introduces a dual certificate condition for atomic norm minimization that guarantees approximate recovery of structured signals from linear measurements, with applications including super-resolution. The condition ensures that peaks in the sparse decomposition of the true signal are close to the support of the atomic decomposition of the solution.
This paper describes a dual certificate condition on a linear measurement operator $A$ (defined on a Hilbert space $\mathcal{H}$ and having finite-dimensional range) which guarantees that an atomic norm minimization, in a certain sense, will be able to approximately recover a structured signal $v_0 \in \mathcal{H}$ from measurements $Av_0$. Put very streamlined, the condition implies that peaks in a sparse decomposition of $v_0$ are close the the support of the atomic decomposition of the solution $v^*$. The condition applies in a relatively general context - in particular, the space $\mathcal{H}$ can be infinite-dimensional. The abstract framework is applied to several concrete examples, one example being super-resolution. In this process, several novel results which are interesting on its own are obtained.