Super-resolution of near-colliding point sources

摘要

We consider the problem of stable recovery of sparse signals of the form F(x) =dΣj=1a_jδ(x ? x_j), x_j ∈ R, a_j ∈ C, from their spectral measurements, known in a bandwidth Ω with absolute error not exceeding ? > 0. We consider the case when at most p ≤ d nodes {x_j} of F form a cluster whose extent is smaller than the Rayleigh limit 1/Ω , while the rest of the nodes is well separated. Provided that ? ? SRF~(?2p+1), where SRF = (ΩΔ)~(?1) and Δ is the minimal separation between the nodes, we show that the minimax error rate for reconstruction of the cluster nodes is of order 1/Ω SRF~(2p?1) ?, while for recovering the corresponding amplitudes {a_j} the rate is of the order SRF~(2p?1) ?. Moreover, the corresponding minimax rates for the recovery of the non-clustered nodes and amplitudes are ?/Ω and ?, 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.