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 = ki=1 aiδti from m samples of its convolution with a window function φ(s−t), of the form y(sj ) = ki=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 mj=1 δ2j ≤ 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.