# Sparse non-negative super-resolution — simplified and stabilised

Eftekhari, A
Tanner, J
Thompson, A
Tyagi, H

13 August 2019

## Journal:

Applied and Computational Harmonic Analysis

## Last Updated:

2020-04-18T09:51:45.887+01:00

## DOI:

10.1016/j.acha.2019.08.004

## abstract:

We consider the problem of non-negative super-resolution, which concerns
reconstructing a non-negative signal x = k
i=1 aiδti from m samples of its
convolution with a window function φ(s−t), of the form y(sj ) = k
i=1 aiφ(sj −ti)+
δj , where δj indicates an inexactness in the sample value. We first show that x is
the unique non-negative measure consistent with the samples, provided the samples
are exact. Moreover, we characterise non-negative solutions xˆ consistent with the
samples within the bound m
j=1 δ2
j ≤ δ2. We show that the integrals of xˆ and x
over (ti − ,ti + ) converge to one another as and δ approach zero and that x and
xˆ are similarly close in the generalised Wasserstein distance. Lastly, we make these
results precise for φ(s − t) Gaussian. The main innovation is that non-negativity is
sufficient to localise point sources and that regularisers such as total variation are
not required in the non-negative setting.

1039749

Submitted

Journal Article