Sparse non-negative super-resolution — simplified and stabilised

We consider the problem of non-negative super-resolution, which concerns reconstructing a non-negative signal x=∑i=1kaiδti from m samples of its convolution with a window function ϕ(s−t), of the form y(sj)=∑i=1kaiϕ(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 ∑j=1mδj2≤δ2. We show that the integrals of xˆ and x over (ti−ϵ,ti+ϵ) converge to one another as ϵ 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.