Super-resolution of near-colliding point sources

Abstract

We consider the problem of stable recovery of sparse signals of the form from their spectral measurements, known in a bandwidth with absolute error not exceeding epsilon>0. We consider the case when at most form a cluster whose extent is smaller than the Rayleigh limit {1over var }, while the rest of the nodes is well separated. Provided that epsilon lessapprox operatorname{SRF}{-2p+1}, where operatorname{SRF}=(var 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 {1over var }operatorname{SRF}{2p-1}epsilon , while for recovering the corresponding amplitudes {aj} the rate is of the order operatorname{SRF}{2p-1}epsilon . Moreover, the corresponding minimax rates for the recovery of the non-clustered nodes and amplitudes are {epsilon over var } 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.