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

Abstract

The convolution of a discrete measure, $x=\sum_{i=1}^ka_iδ_{t_i}$, with a local window function, $ϕ(s-t)$, is a common model for a measurement device whose resolution is substantially lower than that of the objects being observed. Super-resolution concerns localising the point sources $\{a_i,t_i\}_{i=1}^k$ with an accuracy beyond the essential support of $ϕ(s-t)$, typically from $m$ samples $y(s_j)=\sum_{i=1}^k a_iϕ(s_j-t_i)+η_j$, where $η_j$ indicates an inexactness in the sample value. We consider the setting of $x$ being non-negative and seek to characterise all non-negative measures approximately consistent with the samples. We first show that $x$ is the unique non-negative measure consistent with the samples provided the samples are exact, i.e. $η_j=0$, $m\ge 2k+1$ samples are available, and $ϕ(s-t)$ generates a Chebyshev system. This is independent of how close the sample locations are and {\em does not rely on any regulariser beyond non-negativity}; as such, it extends and clarifies the work by Schiebinger et al. and De Castro et al., who achieve the same results but require a total variation regulariser, which we show is unnecessary. Moreover, we characterise non-negative solutions $\hat{x}$ consistent with the samples within the bound $\sum_{j=1}^mη_j^2\le δ^2$. Any such non-negative measure is within ${\mathcal O}(δ^{1/7})$ of the discrete measure $x$ generating the samples in the generalised Wasserstein distance, converging to one another as $δ$ approaches zero. We also show how to make these general results, for windows that form a Chebyshev system, precise for the case of $ϕ(s-t)$ being a Gaussian window. The main innovation of these results is that non-negativity alone is sufficient to localise point sources beyond the essential sensor resolution.