Super-resolution of positive near-colliding point sources

Abstract

Abstract In this paper, we analyze the capacity of super-resolution (SR) of one-dimensional positive sources. In particular, we consider a similar setting as in Batenkov et al. (2020, Inf. Inference, 10, 515–572) and restrict the results to the specific case of super-resolving positive sources. To be more specific, we consider resolving $d$ positive point sources with $p \leqslant d$ nodes closely spaced and forming a cluster, while the rest of the nodes are well separated. Our results show that when the noise level $\epsilon \lesssim \mathrm{SRF}^{-2 p+1}$, where $\mathrm{SRF}=(\varOmega \varDelta )^{-1}$ with $\varOmega $ being the cutoff frequency and $\varDelta $ the minimal separation between the nodes, the minimax error rate for reconstructing the cluster nodes is of order $\frac{1}{\varOmega } \mathrm{SRF}^{2 p-2} \epsilon $, while for recovering the corresponding amplitudes $\{a_j \}$, the rate is of order $\mathrm{SRF}^{2 p-1} \epsilon $. For the non-cluster nodes, the corresponding minimax rates for the recovery of nodes and amplitudes are of order $\frac{\epsilon }{\varOmega }$ and $\epsilon $, respectively. Compared with results for complex sources in Batenkov et al. (2020, Inf. Inference, 10, 515–572), our findings reveal that the positivity of point sources actually does not mitigate the ill-posedness of the SR problem. Although surprising, this fact does not contradict positivity’s significant role in the convex algorithms. In fact, our findings are consistent with existing convex algorithms’ stability results for resolving separation-free positive sources, validating their superior SR capabilities. Moreover, our numerical experiments demonstrate that the Matrix Pencil method perfectly meets the minimax rates for resolving positive sources.