Section: New Results

Axis 2: Sparse non-negative super-resolution - simplified and stabilized

Participant: Hemant Tyagi.

The convolution of a discrete measure, $x={\sum }_{i=1}^k{a}_i{\delta }_{t_i}$, with a local window function, $\phi (s-t)$, 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 with an accuracy beyond the essential support of $\phi (s-t)$, typically from $m$ noisy samples of the convolution output. 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, and $m\ge 2k+1$ samples are available, and $\phi (s-t)$ 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 establish stability results in the setting where the samples are corrupted with noise. The main innovation of these results is that non-negativity alone is sufficient to localise point sources beyond the essential sensor resolution.

This is joint work with Armin Eftekhari (EPFL, Switzerland), Jared Tanner (University of Oxford, UK), Andrew Thompson (National Physical Laboratory, UK), Bogdan Toader (University of Oxford, UK) and was mostly done while Hemant Tyagi was affiliated to the Alan Turing Institute. It has now been published in an international journal [24].