Super-resolution of near-colliding point sources

We consider the problem of stable recovery of sparse signals of the form $F(x)=\sum_{j=1}^d a_jδ(x-x_j),\quad x_j\in\mathbb{R},\;a_j\in\mathbb{C},$ from their spectral measurements, known in a bandwidth $Ω$ with absolute error not exceeding $ε>0$. 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 ${1\overΩ}$, while the rest of the nodes are well separated. Provided that $ε\lessapprox SRF^{-2p+1}$, where $SRF=(ΩΔ)^{-1}$ and $Δ$ is the minimal separation between the nodes, we show that the minimax error rate for reconstruction of the cluster nodes is of order ${1\overΩ}SRF^{2p-1}ε$, while for recovering the corresponding amplitudes $\{a_j\}$ the rate is of the order $SRF^{2p-1}ε$. Moreover, the corresponding minimax rates for the recovery of the non-clustered nodes and amplitudes are ${ε\overΩ}$ and $ε$, 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.

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