# Superresolution without Separation

arXiv: https://arxiv.org/abs/1506.03144 · HTML: https://mlsanity.com/paper/1506.03144/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2015.

## Built on by (most cited first, 6 of 6)

- [Off-the-grid learning of mixtures from a continuous dictionary](https://mlsanity.com/paper/2207.00171/) (2022)
- [Super-Resolution with Structured Motion](https://mlsanity.com/paper/2505.15961/) (2025)
- [A Dual Certificate Approach to Sparsity in Infinite-Width Shallow Neural Networks](https://mlsanity.com/paper/2603.17785/) (2026)
- [The Nonconvex Geometry of Linear Inverse Problems](https://mlsanity.com/paper/2101.02776/) (2021)
- [Characterization of the Basin of Convexity for Multi-Snapshot Spike Deconvolution via Variable Projection](https://mlsanity.com/paper/2607.09593/) (2026)
- [On the Optimisation Landscape of Overparameterised Weighted MMD](https://mlsanity.com/paper/2610.19861/) (2026)

## Abstract (arXiv)

This paper provides a theoretical analysis of diffraction-limited superresolution, demonstrating that arbitrarily close point sources can be resolved in ideal situations. Precisely, we assume that the incoming signal is a linear combination of M shifted copies of a known waveform with unknown shifts and amplitudes, and one only observes a finite collection of evaluations of this signal. We characterize properties of the base waveform such that the exact translations and amplitudes can be recovered from 2M + 1 observations. This recovery is achieved by solving a a weighted version of basis pursuit over a continuous dictionary. Our methods combine classical polynomial interpolation techniques with contemporary tools from compressed sensing.