Super-resolution of near-colliding point sources

We consider the problem of stable recovery of sparse signals of the form $$F(x)=\sum_{j=1}^d a_j\delta(x-x_j),\quad x_j\in\mathbb{R},\;a_j\in\mathbb{C},$$ 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 ${1\over\Omega}$, while the rest of the nodes are well separated. Provided that $\epsilon \lessapprox SRF^{-2p+1}$, 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 ${1\over\Omega}SRF^{2p-1}\epsilon$, while for recovering the corresponding amplitudes $\{a_j\}$ the rate is of the order $SRF^{2p-1}\epsilon$. Moreover, the corresponding minimax rates for the recovery of the non-clustered nodes and amplitudes are ${\epsilon\over\Omega}$ 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.