Super-resolution of near-colliding point sources

Abstract

We consider the problem of stable recovery of sparse signals of the form $$F(x)=\sum _{j=1}^d{a}_j\delta (x-x_j),x_j\in \mathbb{R},a_j\in \mathbb{C},$$ from their spectral measurements, known in a bandwidth $\Omega$ with absolute error not exceeding $\epsilon >0$. We consider the case when at most $p\le d$ nodes $\{x_j\}$ of $F$ form a cluster whose extent is smaller than the Rayleigh limit $\frac{1}{\Omega }$, while the rest of the nodes are well separated. Provided that $\epsilon ⪅SR{F}^{-2p+1}$, where $SRF=(\Omega \Delta {)}^{-1}$ and $\Delta$ is the minimal separation between the nodes, we show that the minimax error rate for reconstruction of the cluster nodes is of order $\frac{1}{\Omega }SR{F}^{2p-1}\epsilon$, while for recovering the corresponding amplitudes $\{a_j\}$ the rate is of the order $SR{F}^{2p-1}\epsilon$. Moreover, the corresponding minimax rates for the recovery of the non-clustered nodes and amplitudes are $\frac{\epsilon }{\Omega }$ and $\epsilon$, respectively. These results suggest that stable super-resolution is possible in much more general situations than previously thought. Our numerical experiments show that the well-known Matrix Pencil method achieves the above accuracy bounds.