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 equation*F(x)=_j=1^d a_jδ(x-x_j), x_jR, a_jC, equation* from their spectral measurements, known in a bandwidth with absolute error not exceeding ε>0 . We consider the case when at most pd nodes {x_j} of F form a cluster whose extent is smaller than the Rayleigh limit 1, while the rest of the nodes is well separated. Provided that ε SRF^-2p+1, where SRF=()^-1 and is the minimal separation between the nodes, we show that the minimax error rate for reconstruction of the cluster nodes is of order 1SRF^2p-1ε , while for recovering the corresponding amplitudes {a_j} the rate is of the order SRF^2p-1ε . Moreover, the corresponding minimax rates for the recovery of the non-clustered nodes and amplitudes are ε 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.