Super-resolution of near-colliding point sources. (11th May 2020) - Batenkov, Dmitry; Goldman, Gil; Yomdin, Yosef
Abstract: We consider the problem of stable recovery of sparse signals of the form $$\begin{equation*}F(x)=\sum_{j=1}^d a_jδ(x-x_j), \quad x_j\in\mathbb{R}, \;a_j\in\mathbb{C}, \end{equation*}$$ from their spectral measurements, known in a bandwidth $\varOmega $ with absolute error not exceeding $ε>0$ . We consider the case when at most $p\leqslant d$ nodes $\{x_j\}$ of $F$ form a cluster whose extent is smaller than the Rayleigh limit ${1\over \varOmega }$, while the rest of the nodes is well separated. Provided that $ε \lessapprox \operatorname{SRF}^{-2p+1}$, where $\operatorname{SRF}=(\varOmega \varDelta )^{-1}$ and $\varDelta $ 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 \varOmega }\operatorname{SRF}^{2p-1}ε $, while for recovering the corresponding amplitudes $\{a_j\}$ the rate is of the order $\operatorname{SRF}^{2p-1}ε $ . Moreover, the corresponding minimax rates for the recovery of the non-clustered nodes and amplitudes are ${ε \over \varOmega }$ 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.