Sparse non-negative super-resolution - simplified and stabilised
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The convolution of a discrete measure, x=∑_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)=∑_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> 2k+1 samples are available, and ϕ(s-t) generates a Chebyshev system. This is independent of how close the sample locations are and 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 x̂ consistent with the samples within the bound ∑_j=1^mη_j^2<δ^2. Any such non-negative measure is within 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.
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