The recoverability limit for superresolution via sparsity

TL;DR

This paper establishes bounds on the error rate for recovering sparse signals from Fourier data, revealing how superresolution limits affect robustness to noise and providing insights into the problem's fundamental difficulty.

Contribution

It provides the first bounds on the minimax error rate in superresolution sparse recovery, linking superresolution factor to noise robustness and introducing new bounds on Fourier matrix singular values.

Findings

Error bounds scale as (SRF)^{2k-1} times noise level

Bounds depend on restricted isometry constants and σ-spark

New bounds on Fourier matrix singular values are derived

Abstract

We consider the problem of robustly recovering a $k$-sparse coefficient vector from the Fourier series that it generates, restricted to the interval $[- \Omega, \Omega]$. The difficulty of this problem is linked to the superresolution factor SRF, equal to the ratio of the Rayleigh length (inverse of $\Omega$) by the spacing of the grid supporting the sparse vector. In the presence of additive deterministic noise of norm $\sigma$, we show upper and lower bounds on the minimax error rate that both scale like $(SRF)^{2k-1} \sigma$, providing a partial answer to a question posed by Donoho in 1992. The scaling arises from comparing the noise level to a restricted isometry constant at sparsity $2k$, or equivalently from comparing $2k$ to the so-called $\sigma$-spark of the Fourier system. The proof involves new bounds on the singular values of restricted Fourier matrices, obtained in part from old techniques in complex analysis.