Super-resolution of near-colliding point sources

2019/04/19 by Batenkov, Dmitry, Goldman, Gil, Yomdin, Yosef · 3 citations
#42A10 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1904.09186

Abstract

We consider the problem of stable recovery of sparse signals of the form F(x)=∑_j=1^d a_jδ(x-x_j), x_j∈ℝ, a_j∈ℂ, from their spectral measurements, known in a bandwidth Ω with absolute error not exceeding ε>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\overΩ, while the rest of the nodes are well separated. Provided that ε\lessapprox 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\overΩSRF^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\overΩ 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.

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